Line data Source code
1 0 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : #pragma once
5 :
6 : #include "DataStructures/Tensor/TypeAliases.hpp"
7 : #include "Utilities/Gsl.hpp"
8 :
9 : /// \cond
10 : class DataVector;
11 : namespace Frame {
12 : struct Distorted;
13 : struct Grid;
14 : } // namespace Frame
15 : /// \endcond
16 :
17 : namespace control_system::size {
18 :
19 : /*!
20 : * \brief Computes the derivative of the comoving characteristic speed
21 : * with respect to the size map parameter.
22 : *
23 : * \param result the derivative of the comoving char speed
24 : * \f$d v_c/d\lambda_{00}\f$, which is computed here using
25 : * Eq. (\f$\ref{eq:result}\f$).
26 : * \param lambda_00 the map parameter \f$\lambda_{00}\f$. This is the usual
27 : * spherical harmonic coefficient, not a Spherepack value.
28 : * \param dt_lambda_00 the time derivative of the map parameter
29 : * \param horizon_00 the average coefficient of the horizon \f$\hat{S}_{00}\f$.
30 : * This is the usual spherical harmonic coefficient, not a Spherepack
31 : * value.
32 : * \param dt_horizon_00 the time derivative of horizon_00
33 : * \param grid_frame_excision_sphere_radius radius of the excision boundary
34 : * in the grid frame, \f$r_{\mathrm{EB}}\f$.
35 : * \param excision_rhat the direction cosine \f$\xi_\hat{i}\f$. Not a
36 : * spacetime tensor: it is raised/lowered with \f$\delta_{ij}\f$
37 : * \param excision_normal_one_form the unnormalized one-form
38 : * \f$\hat{s}_\hat{i}\f$
39 : * \param one_over_excision_normal_one_form_norm one over the norm of the
40 : * one-form \f$a\f$
41 : * \param distorted_components_of_grid_shift the quantity
42 : * \f$\beta^i \frac{\partial x^\hat{i}}{\partial x_i}\f$
43 : * evaluated on the excision boundary. This is not the shift in
44 : * the distorted frame.
45 : * \param inverse_spatial_metric_on_excision_boundary metric in
46 : * the distorted frame.
47 : * \param spatial_christoffel_second_kind the Christoffel symbols
48 : * \f$\Gamma^\hat{k}_{\hat{i}\hat{j}}\f$
49 : * \param deriv_lapse the spatial derivative of the lapse
50 : * \f$\partial_\hat{i} \alpha\f$
51 : * \param deriv_of_distorted_shift the spatial derivative of the shift in the
52 : * distorted frame
53 : * \f$\partial_\hat{j} \hat{\beta}^\hat{i}\f$. This is not the
54 : * derivative of distorted_components_of_grid_shift.
55 : * \param inverse_jacobian_grid_to_distorted the quantity
56 : * \f$J^i_\hat{k}=\partial_\hat{k} x^i\f$,
57 : * where \f$x^i\f$ are the grid frame coordinates and
58 : * \f$x^{\hat k}\f$ are the distorted frame coordinates.
59 : * ## Background
60 : *
61 : * The characteristic speed on the excision boundary is
62 : * \f{align}
63 : * v &= -\alpha + n_i\beta^i
64 : * \f}
65 : * where \f$\alpha\f$ is the lapse (invariant under frame transformations),
66 : * \f$\beta^i\f$ is the grid-frame shift, and \f$n_i\f$ is the metric-normalized
67 : * **outward-pointing** (i.e. pointing out of the black hole,
68 : * toward larger radius)
69 : * normal one-form to the excision boundary in the grid frame.
70 : * (Note that the usual expression for the characteristic speed, as in
71 : * eq. 87 of \cite Hemberger2012jz, has
72 : * a minus sign and defines \f$n_i\f$ as the inward-pointing (i.e. out of the
73 : * computational domain) normal; here
74 : * we have a plus sign and we define \f$n_i\f$ as outward-pointing because
75 : * the outward-pointing normal is passed into comoving_char_speed_derivative.)
76 : *
77 : * The size/shape map at the excision boundary is given by Eq. 72 of
78 : * \cite Hemberger2012jz :
79 : * \f{align}
80 : * \hat{x}^i &= \frac{x^i}{r_{\mathrm{EB}}}
81 : * \left(1 - \lambda_{00} Y_{00}
82 : * -\sum_{\ell>0} Y_{\ell m} \lambda_{\ell m}\right),
83 : * \label{eq:map}
84 : * \f}
85 : * where \f$\hat{x}^i\f$ are the distorted-frame coordinates and \f$x^i\f$
86 : * are the grid-frame coordinates, and where we have separated the \f$\ell=0\f$
87 : * piece from the sum.
88 : * Here \f$Y_{\ell m}\f$ are
89 : * spherical harmonics, \f$\lambda_{\ell m}\f$ are
90 : * the map parameters, and \f$r_{\mathrm{EB}}\f$ is the radius of the
91 : * excision boundary in the
92 : * grid frame (where the excision boundary is a sphere). The final term with
93 : * the sum over $\ell>0$ is independent of \f$\lambda_{00}\f$,
94 : * and will not be important
95 : * because below we will be differentiating the map with respect
96 : * to \f$\lambda_{00}\f$.
97 : *
98 : * The comoving characteristic speed is given by rewriting Eq. 98
99 : * of \cite Hemberger2012jz in terms of the distorted-frame shift:
100 : * \f{align}
101 : * v_c &= -\alpha +\hat{n}_\hat{i}\hat{\beta}^\hat{i}
102 : * - Y_{00} \hat{n}_{\hat i} \xi^{\hat i}
103 : * \left[ \dot{\hat{S}}_{00} (\lambda_{00}
104 : * - r_{\mathrm{EB}}/Y_{00}) / \hat{S}_{00}
105 : * + \frac{1}{Y_{00}} \sum_{\ell>0} Y_{\ell m} \dot{\lambda}_{\ell m}
106 : * \right], \\
107 : * &= -\alpha +\hat{n}_\hat{i}\beta^\hat{i}
108 : * - Y_{00} \hat{n}_{\hat i} \xi^{\hat i}
109 : * \left[ \dot{\hat{S}}_{00} (\lambda_{00}
110 : * - r_{\mathrm{EB}}/Y_{00}) / \hat{S}_{00}
111 : * -\dot{\lambda}_{00} \right], \label{eq:comovingspeed}
112 : * \f}
113 : * where in the last line we have rewritten $\hat{\beta}^\hat{i}$
114 : * in terms of $\beta^\hat{i}$ (see Eq. (\f$\ref{eq:framecompsshiftdef}\f$)
115 : * below) and we have substituted
116 : * the time derivative of Eq. (\f$\ref{eq:map}\f$).
117 : * Here \f$\dot{\lambda}_{00}\f$ is the time derivative of
118 : * \f$\lambda_{00}\f$, and
119 : * \f$\hat{S}_{00}\f$ is the constant spherical-harmonic coefficient of the
120 : * horizon and \f$\dot{\hat{S}}_{00}\f$ is its time derivative.
121 : * The symbol \f$\xi^{\hat i}\f$ is
122 : * a direction cosine, i.e. \f$x^i/r_{\mathrm{EB}}\f$ evaluated on the
123 : * excision boundary, which is the same as
124 : * \f$\hat{x}^i/\hat{r}_{\mathrm{EB}}\f$ evaluated on the excision boundary
125 : * because the size and shape maps preserve angles. Note that
126 : * \f$r_{\mathrm{EB}}\f$ is a constant but \f$\hat{r}_{\mathrm{EB}}\f$ is
127 : * a function of angles. Note also
128 : * that \f$\xi^{\hat i}\f$ is **not** a vector; it
129 : * is a coordinate quantity. In particular,
130 : * the lower-index \f$\xi_{\hat i}\f$ is \f$\delta_{ij}x^j/r_{\mathrm{EB}}\f$.
131 : * The non-vectorness of \f$\xi^{\hat i}\f$ (and of \f$x^i\f$ itself
132 : * in Eq. (\f$\ref{eq:map}\f$)) might cause some confusion when using the
133 : * Einstein summation convention; we attempt to alleviate that confusion by
134 : * never using the lower-index \f$\xi_{\hat i}\f$ and by keeping
135 : * \f$\delta_{ij}\f$ in formulas below.
136 : * The normal
137 : * \f$\hat{n}_\hat{i}\f$ is the same as $n_i$
138 : * transformed into the distorted frame, that is
139 : * \f$\hat{n}_\hat{i} = n_j \partial x^j/\partial\hat{x}^\hat{i}\f$.
140 : * We have put a hat on \f$\hat{n}\f$ in addition to putting a hat on
141 : * its index
142 : * (despite the usual convention that tensors have
143 : * decorations on indices and not on the tensors themselves)
144 : * to reduce later ambiguities
145 : * in notation that arise because
146 : * Eq. (\f$\ref{eq:map}\f$) has the same index on both sides of the equation
147 : * and because \f$\xi^{\hat i}\f$ and \f$x^i\f$ are not tensors.
148 : * The quantity \f$\beta^\hat{i}\f$ in Eq. (\f$\ref{eq:comovingspeed}\f$)
149 : * is the distorted-frame
150 : * component of the grid-frame shift, defined by
151 : * \f{align}
152 : * \beta^\hat{i} &= \beta^i \frac{\partial \hat{x}^\hat{i}}{\partial x^i}.
153 : * \label{eq:shiftyquantity}
154 : * \f}
155 : * This is **not** the shift in the distorted frame \f$\hat{\beta}^\hat{i}\f$,
156 : * because the shift does
157 : * not transform like a spatial tensor under the maps.
158 : *
159 : * If the comoving characteristic speed \f$v_c\f$ is negative and remains
160 : * negative forever, then size control will fail. Therefore \f$v_c\f$ is
161 : * used in making decisions in size control. We care about
162 : * \f$d v_c/d\lambda_{00}\f$ because the sign of that quantity tells us
163 : * whether \f$v_c\f$ will increase or decrease when we increase or decrease
164 : * the map parameter \f$\lambda_{00}\f$; this information will be used to
165 : * decide whether to transition between different size control states.
166 : *
167 : * ## Derivation of derivative of comoving characteristic speed
168 : *
169 : * This function computes
170 : * \f$d v_c/d\lambda_{00}\f$ on the excision boundary, where the total
171 : * derivative means that all other map parameters
172 : * (like \f$\lambda_{\ell m}\f$ for \f$\ell>0\f$) are held fixed, and the
173 : * coordinates of the excision boundary (the grid coordinates) are held fixed.
174 : * We also hold fixed \f$\dot{\lambda}_{00}\f$ because we are interested
175 : * in how $v_c$ changes from a configuration with a given
176 : * \f$\lambda_{00}\f$ and \f$\dot{\lambda}_{00}\sim 0\f$ to another
177 : * configuration with a different nearby \f$\lambda_{00}\f$ and also with
178 : * \f$\dot{\lambda}_{00}\sim 0\f$.
179 : *
180 : * Here we derive an expression for \f$d v_c/d\lambda_{00}\f$.
181 : * This expression will be
182 : * complicated, mostly because of the normals \f$\hat{n}_\hat{i}\f$ that appear
183 : * in Eq. (\f$\ref{eq:comovingspeed}\f$) and because of the Jacobians.
184 : *
185 : * ### Derivative of the Jacobian
186 : *
187 : * First, note that by differentiating Eq. (\f$\ref{eq:map}\f$) we obtain
188 : * \f{align}
189 : * \frac{d\hat{x}^i}{d\lambda_{00}}
190 : * &= - \frac{x^i Y_{00}}{r_{\mathrm{EB}}} \\
191 : * &= -\xi^i Y_{00},
192 : * \f}
193 : * where the last line follows from the definition of the direction cosines.
194 : *
195 : * The Jacobian of the map is
196 : * \f{align}
197 : * \frac{\partial \hat{x}^i}{\partial x^j}
198 : * &= (1 - \lambda_{00} Y_{00}/r_{\mathrm{EB}} + B) \delta^i_j
199 : * + x^i \frac{\partial B}{\partial x^j},
200 : * \f}
201 : * where \f$B\f$ represents the term with the sum over \f$\ell>0\f$
202 : * in Eq. (\f$\ref{eq:map}\f$); this term is independent
203 : * of \f$\lambda_{00}\f$.
204 : * Therefore, we have
205 : * \f{align}
206 : * \frac{d}{d\lambda_{00}}\frac{\partial \hat{x}^i}{\partial x^j} &=
207 : * -\frac{Y_{00}}{r_{\mathrm{EB}}} \delta^i_j.
208 : * \label{eq:derivjacobian}
209 : * \f}
210 : *
211 : * But we want the derivative of the inverse Jacobian, not the forward
212 : * Jacobian. By taking the derivative of the identity
213 : * \f{align}
214 : * \frac{\partial \hat{x}^\hat{i}}{\partial x^k}
215 : * \frac{\partial x^k}{\partial \hat{x}^\hat{j}} &= \delta^\hat{i}_\hat{j}
216 : * \f}
217 : * and by using Eq. (\f$\ref{eq:derivjacobian}\f$) we can derive
218 : * \f{align}
219 : * \frac{d}{d\lambda_{00}}\frac{\partial x^i}{\partial \hat{x}^j} &=
220 : * +\frac{Y_{00}}{r_{\mathrm{EB}}}
221 : * \frac{\partial x^i}{\partial \hat{x}^k}
222 : * \frac{\partial x^k}{\partial \hat{x}^j}.
223 : * \label{eq:strangederivjacobian}
224 : * \f}
225 : * Note that the right-hand side of Eq. (\f$\ref{eq:strangederivjacobian}\f$)
226 : * has two inverse Jacobians contracted with each other, which is not
227 : * the same as \f$\delta^i_j\f$.
228 : *
229 : * ### Derivative of a function of space
230 : *
231 : * Assume we have an arbitrary function of space \f$f(\hat{x}^i)\f$.
232 : * Here we treat \f$f\f$ as a function of the distorted-frame
233 : * coordinates \f$\hat{x}^i\f$ and not a function of the grid-frame
234 : * coordinates. This is because we consider the metric functions to be defined
235 : * in the inertial frame (and equivalently for our purposes the functions are
236 : * defined in the distorted frame because the distorted-to-inertial map
237 : * is independent of \f$\lambda_{00}\f$), and we consider \f$\lambda_{00}\f$
238 : * a parameter in a map that moves the grid with respect to these
239 : * distorted-frame metric functions.
240 : * The derivative of \f$f\f$ can be written
241 : * \f{align}
242 : * \frac{d}{d\lambda_{00}}f &= \frac{\partial f}{\partial \hat{x}^i}
243 : * \frac{d \hat{x}^i}{d\lambda_{00}}\\
244 : * &= -\xi^\hat{i} Y_{00} \frac{\partial f}{\partial \hat{x}^i}.
245 : * \label{eq:derivf}
246 : * \f}
247 : * This is how we will evaluate derivatives of metric functions like
248 : * the lapse.
249 : *
250 : * ### Derivative of the distorted-frame components of the grid-frame shift.
251 : *
252 : * To differentiate the quantity defined by Eq. (\f$\ref{eq:shiftyquantity}\f$)
253 : * note that
254 : * \f{align}
255 : * \beta^\hat{i} &=
256 : * \beta^i \frac{\partial \hat{x}^\hat{i}}{\partial x^i} \\
257 : * &= \hat{\beta}^\hat{i} + \frac{\partial \hat{x}^\hat{i}}{\partial t},
258 : * \label{eq:framecompsshiftdef}
259 : * \f}
260 : * where \f$\hat{\beta}^\hat{i} \equiv \alpha^2 g^{\hat{0}\hat{i}}\f$ is
261 : * the shift in the distorted frame.
262 : * From the map, Eq. (\f$\ref{eq:map}\f$), we see that
263 : * \f{align}
264 : * \frac{d}{d\lambda_{00}} \frac{\partial \hat{x}^\hat{i}}{\partial t} &=0,
265 : * \f}
266 : * because there is no remaining \f$\lambda_{00}\f$ in
267 : * \f$\frac{\partial \hat{x}^\hat{i}}{\partial t}\f$.
268 : * So
269 : * \f{align}
270 : * \frac{d}{d\lambda_{00}}\beta^\hat{i} &=
271 : * \frac{d}{d\lambda_{00}} \hat{\beta}^\hat{i} \\
272 : * &= -\xi^\hat{j} Y_{00} \partial_\hat{j} \hat{\beta}^\hat{i},
273 : * \f}
274 : * where we have used Eq. (\f$\ref{eq:derivf}\f$) in the last line.
275 : * Note that we cannot use Eq. (\f$\ref{eq:derivf}\f$) on
276 : * \f$\beta^\hat{i}\f$ directly,
277 : * because \f$\beta^\hat{i}\f$ depends in a complicated
278 : * way on the grid-to-distorted map. In particular, we will be evaluating
279 : * \f$\partial_\hat{j} \hat{\beta}^\hat{i}\f$ numerically, and numerical
280 : * spatial derivatives \f$\partial_\hat{j} \hat{\beta}^\hat{i}\f$ are not
281 : * the same as numerical spatial derivatives
282 : * \f$\partial_\hat{j} \beta^\hat{i}\f$.
283 : *
284 : * ### Derivative of the normal one-form
285 : *
286 : * The normal to the surface is the most complicated expression in
287 : * Eq. (\f$\ref{eq:comovingspeed}\f$), because of how it depends on the
288 : * map and on the metric.
289 : * The grid-frame un-normalized outward-pointing one-form
290 : * to the excision boundary is
291 : * \f{align}
292 : * s_i &= \xi^j \delta_{ij},
293 : * \f}
294 : * because the excision boundary is a sphere of fixed radius in the
295 : * grid frame. Therefore
296 : * \f$s_i\f$ doesn't depend on \f$\lambda_{00}\f$.
297 : *
298 : * The normalized one-form \f$\hat{n}_\hat{i}\f$ is given by
299 : * \f{align}
300 : * \hat{n}_\hat{i} &= \frac{\hat{s}_{\hat i}}{a},
301 : * \f}
302 : * where
303 : * \f{align}
304 : * \hat{s}_{\hat i} &= s_i \frac{\partial x^i}{\partial \hat{x}^{\hat i}},\\
305 : * a^2 &= \hat{s}_{\hat i} \hat{s}_{\hat j} \gamma^{\hat{i} \hat{j}}.
306 : * \f}
307 : * Here \f$\gamma^{\hat{i} \hat{j}}\f$ is the inverse 3-metric in the
308 : * distorted frame. Again, to avoid ambiguity later,
309 : * we have put hats on \f$n\f$ and \f$s\f$, despite
310 : * the usual convention that when transforming tensors one puts
311 : * hats on the indices and not on the tensors.
312 : *
313 : * Now
314 : * \f{align}
315 : * \frac{d}{d\lambda_{00}} \hat{s}_{\hat i} &=
316 : * \frac{Y_{00}}{r_{\mathrm{EB}}}
317 : * \hat{s}_k\frac{\partial x^k}{\partial \hat{x}^\hat{i}}, \\
318 : * \frac{d}{d\lambda_{00}} a^2 &= 2 \frac{Y_{00}}{r_{\mathrm{EB}}}
319 : * \hat{s}_k\frac{\partial x^k}{\partial \hat{x}^\hat{i}}
320 : * \hat{s}_{\hat j} \gamma^{\hat{i} \hat{j}}
321 : * + \hat{s}_{\hat i} \hat{s}_{\hat j}
322 : * \gamma^{\hat{i} \hat{k}} \gamma^{\hat{j} \hat{l}}
323 : * \xi^\hat{m} Y_{00} \partial_{\hat m} \gamma_{\hat{k} \hat{l}}.
324 : * \f}
325 : * Here we have used Eq. (\f$\ref{eq:strangederivjacobian}\f$) to differentiate
326 : * the Jacobian, and Eq. (\f$\ref{eq:derivf}\f$) to differentiate the 3-metric.
327 : * We have also refrained from raising and lowering indices
328 : * on \f$\hat{n}_\hat{i}\f$, \f$\hat{s}_\hat{i}\f$, and \f$\xi^\hat{i}\f$
329 : * to alleviate potential confusion over whether to raise or lower using
330 : * \f$\gamma_{\hat{i} \hat{j}}\f$ or using \f$\delta_{\hat{i}\hat{j}}\f$.
331 : * The factor \f$\hat{s}_k \partial x^k/\partial \hat{x}^\hat{i}\f$
332 : * is unusal and is not a tensor
333 : * (\f$\hat{s}_k\f$ is a tensor but the Jacobian it is being multiplied by
334 : * is the inverse of the one that would transform it into a different frame);
335 : * this factor arises because some quantities being differentiated are
336 : * not tensors.
337 : *
338 : * Given the above, the derivative of the normalized normal one-form is
339 : * \f{align}
340 : * \frac{d}{d\lambda_{00}} \hat{n}_{\hat i} &=
341 : * \frac{1}{a}\frac{d}{d\lambda_{00}} \hat{s}_{\hat i}
342 : * - \hat{s}_{\hat i} \frac{1}{2a^3}
343 : * \frac{d}{d\lambda_{00}} a^2\\
344 : * &=
345 : * \hat{s}_i\frac{Y_{00}}{a r_{\mathrm{EB}}}
346 : * \frac{\partial x^i}{\partial \hat{x}^\hat{i}}
347 : * - \hat{s}_{\hat i} \frac{1}{a^3} \hat{s}_i\frac{Y_{00}}{r_{\mathrm{EB}}}
348 : * \frac{\partial x^i}{\partial \hat{x}^\hat{k}}
349 : * \hat{s}_{\hat j} \gamma^{\hat{k} \hat{j}}
350 : * - \hat{s}_{\hat i} \frac{Y_{00}}{2a^3} \hat{s}_{\hat p}
351 : * \hat{s}_{\hat j} \gamma^{\hat{p} \hat{k}}
352 : * \gamma^{\hat{j} \hat{l}}
353 : * \xi^\hat{m} \partial_{\hat m} \gamma_{\hat{k} \hat{l}} \\
354 : * &=
355 : * \hat{n}_i
356 : * \frac{\partial x^i}{\partial \hat{x}^\hat{k}}
357 : * \frac{Y_{00}}{r_{\mathrm{EB}}}
358 : * (\delta^\hat{k}_\hat{i} - \hat{n}^\hat{k} \hat{n}_\hat{i})
359 : * - \hat{s}_{\hat i} \frac{Y_{00}}{2a^3} \hat{s}_{\hat p}
360 : * \hat{s}_{\hat j} \gamma^{\hat{p} \hat{k}}
361 : * \gamma^{\hat{j} \hat{l}}
362 : * \xi^\hat{m} \partial_{\hat m} \gamma_{\hat{k} \hat{l}}
363 : * \label{eq:dnormal} \\
364 : * &=
365 : * \hat{n}_i
366 : * \frac{\partial x^i}{\partial \hat{x}^\hat{k}}
367 : * \frac{Y_{00}}{r_{\mathrm{EB}}}
368 : * (\delta^\hat{k}_\hat{i} - \hat{n}^\hat{k} \hat{n}_\hat{i})
369 : * - Y_{00} \hat{n}_{\hat i} \hat{n}_{\hat p}
370 : * \hat{n}_{\hat j} \gamma^{\hat{p} \hat{k}}
371 : * \xi^\hat{m} \Gamma^\hat{j}_{\hat{k} \hat{m}}
372 : * \label{eq:dnormalgamma},
373 : * \f}
374 : * where we have eliminated \f$\hat{s}_{\hat i}\f$ and \f$a\f$ in favor
375 : * of \f$\hat{n}_{\hat i}\f$
376 : * and we have substituted 3-Christoffel symbols for
377 : * spatial derivatives of the 3-metric (and the factor of 2 on the penultimate
378 : * line has been absorbed into the 3-Christoffel symbol on the last line).
379 : * Note that the last term in Eq.
380 : * (\f$\ref{eq:dnormalgamma}\f$) could also be derived by differentiating
381 : * \f$\hat{n}_\hat{i}\hat{n}_\hat{j}\gamma^{\hat{i}\hat{j}}=1\f$.
382 : * The first term in Eq. (\f$\ref{eq:dnormalgamma}\f$) is strange because
383 : * the inverse Jacobian (as opposed to the forward Jacobian) is contracted
384 : * with \f$\hat{n}_i\f$, so that is not a tensor transformation.
385 : *
386 : * We can now differentiate Eq. (\f$\ref{eq:comovingspeed}\f$) to obtain
387 : * \f{align}
388 : * \frac{d}{d\lambda_{00}} v_c &=
389 : * \xi^\hat{i} Y_{00} \partial_\hat{i} \alpha
390 : * +\left[ \beta^\hat{i}
391 : * - Y_{00} \xi^\hat{i} \dot{\hat{S}}_{00} (\lambda_{00}
392 : * - r_{\mathrm{EB}}/Y_{00}) / \hat{S}_{00}
393 : * + Y_{00} \xi^\hat{i}\dot{\lambda}_{00} \right]
394 : * \frac{d}{d\lambda_{00}} \hat{n}_{\hat i} \nonumber \\
395 : * &- \hat{n}_{\hat i} \xi^\hat{j} Y_{00} \partial_\hat{j} \hat{\beta}^\hat{i}
396 : * - Y_{00} \hat{n}_{\hat i} \xi^{\hat i} \dot{\hat{S}}_{00}/\hat{S}_{00},
397 : * \label{eq:result}
398 : * \f}
399 : * where \f$\frac{d}{d\lambda_{00}} \hat{n}_{\hat i}\f$ is given by
400 : * Eq. (\f$\ref{eq:dnormalgamma}\f$).
401 : */
402 1 : void comoving_char_speed_derivative(
403 : gsl::not_null<Scalar<DataVector>*> result, double lambda_00,
404 : double dt_lambda_00, double horizon_00, double dt_horizon_00,
405 : double grid_frame_excision_sphere_radius,
406 : const tnsr::i<DataVector, 3, Frame::Distorted>& excision_rhat,
407 : const tnsr::i<DataVector, 3, Frame::Distorted>& excision_normal_one_form,
408 : const Scalar<DataVector>& one_over_excision_normal_one_form_norm,
409 : const tnsr::I<DataVector, 3, Frame::Distorted>&
410 : distorted_components_of_grid_shift,
411 : const tnsr::II<DataVector, 3, Frame::Distorted>&
412 : inverse_spatial_metric_on_excision_boundary,
413 : const tnsr::Ijj<DataVector, 3, Frame::Distorted>&
414 : spatial_christoffel_second_kind,
415 : const tnsr::i<DataVector, 3, Frame::Distorted>& deriv_lapse,
416 : const tnsr::iJ<DataVector, 3, Frame::Distorted>& deriv_of_distorted_shift,
417 : const InverseJacobian<DataVector, 3, Frame::Grid, Frame::Distorted>&
418 : inverse_jacobian_grid_to_distorted);
419 : } // namespace control_system::size
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