Line data Source code
1 0 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : #pragma once
5 :
6 : #include <cstddef>
7 :
8 : #include "DataStructures/Tensor/TypeAliases.hpp"
9 :
10 : /// \cond
11 : namespace gsl {
12 : template <typename>
13 : struct not_null;
14 : } // namespace gsl
15 : /// \endcond
16 :
17 : namespace gr {
18 :
19 : /// @{
20 : /*!
21 : * \ingroup GeneralRelativityGroup
22 : * \brief Compute projection operator onto an interface
23 : *
24 : * \details Returns the operator \f$P^{ij} = \gamma^{ij} - n^i n^j\f$,
25 : * where \f$\gamma^{ij}\f$ is the inverse spatial metric, and
26 : * \f$n^i\f$ is the normal vector to the interface in question.
27 : *
28 : */
29 : template <typename DataType, size_t VolumeDim, typename Frame>
30 1 : tnsr::II<DataType, VolumeDim, Frame> transverse_projection_operator(
31 : const tnsr::II<DataType, VolumeDim, Frame>& inverse_spatial_metric,
32 : const tnsr::I<DataType, VolumeDim, Frame>& normal_vector);
33 :
34 : template <typename DataType, size_t VolumeDim, typename Frame>
35 1 : void transverse_projection_operator(
36 : gsl::not_null<tnsr::II<DataType, VolumeDim, Frame>*> projection_tensor,
37 : const tnsr::II<DataType, VolumeDim, Frame>& inverse_spatial_metric,
38 : const tnsr::I<DataType, VolumeDim, Frame>& normal_vector);
39 : /// @}
40 :
41 : /*!
42 : * \ingroup GeneralRelativityGroup
43 : * \brief Compute projection operator onto an interface
44 : *
45 : * \details Returns the operator \f$P_{ij} = \gamma_{ij} - n_i n_j\f$,
46 : * where \f$ \gamma_{ij}\f$ is the spatial metric, and \f$ n_i\f$ is
47 : * the normal one-form to the interface in question.
48 : */
49 : template <typename DataType, size_t VolumeDim, typename Frame>
50 1 : tnsr::ii<DataType, VolumeDim, Frame> transverse_projection_operator(
51 : const tnsr::ii<DataType, VolumeDim, Frame>& spatial_metric,
52 : const tnsr::i<DataType, VolumeDim, Frame>& normal_one_form);
53 :
54 : /*!
55 : * \ingroup GeneralRelativityGroup
56 : * \brief Compute projection operator onto an interface
57 : *
58 : * \details Returns the operator \f$P_{ij} = \gamma_{ij} - n_i n_j\f$,
59 : * where \f$ \gamma_{ij}\f$ is the spatial metric, and \f$ n_i\f$ is
60 : * the normal one-form to the interface in question.
61 : */
62 : template <typename DataType, size_t VolumeDim, typename Frame>
63 1 : void transverse_projection_operator(
64 : gsl::not_null<tnsr::ii<DataType, VolumeDim, Frame>*> projection_tensor,
65 : const tnsr::ii<DataType, VolumeDim, Frame>& spatial_metric,
66 : const tnsr::i<DataType, VolumeDim, Frame>& normal_one_form);
67 :
68 : /*!
69 : * \ingroup GeneralRelativityGroup
70 : * \brief Compute projection operator onto an interface
71 : *
72 : * \details Returns the operator \f$P^{i}_{j} = \delta^{i}_{j} - n^i n_j\f$,
73 : * where \f$n^i\f$ and \f$n_i\f$ are the normal vector and normal one-form
74 : * to the interface in question.
75 : */
76 : template <typename DataType, size_t VolumeDim, typename Frame>
77 1 : tnsr::Ij<DataType, VolumeDim, Frame> transverse_projection_operator(
78 : const tnsr::I<DataType, VolumeDim, Frame>& normal_vector,
79 : const tnsr::i<DataType, VolumeDim, Frame>& normal_one_form);
80 :
81 : /*!
82 : * \ingroup GeneralRelativityGroup
83 : * \brief Compute projection operator onto an interface
84 : *
85 : * \details Returns the operator \f$P^{i}_{j} = \delta^{i}_{j} - n^i n_j\f$,
86 : * where \f$n^i\f$ and \f$n_i\f$ are the normal vector and normal one-form
87 : * to the interface in question.
88 : */
89 : template <typename DataType, size_t VolumeDim, typename Frame>
90 1 : void transverse_projection_operator(
91 : gsl::not_null<tnsr::Ij<DataType, VolumeDim, Frame>*> projection_tensor,
92 : const tnsr::I<DataType, VolumeDim, Frame>& normal_vector,
93 : const tnsr::i<DataType, VolumeDim, Frame>& normal_one_form);
94 :
95 : /// @{
96 : /*!
97 : * \ingroup GeneralRelativityGroup
98 : * \brief Compute spacetime projection operator onto an interface
99 : *
100 : * \details Consider a \f$d-1\f$-dimensional surface \f$S\f$ in a
101 : * \f$d\f$-dimensional spatial hypersurface \f$\Sigma\f$. Let \f$s_a\f$
102 : * be the unit spacelike one-form orthogonal to \f$S\f$ in \f$\Sigma\f$,
103 : * and \f$n_a\f$ be the timelike unit vector orthogonal to \f$\Sigma\f$.
104 : * This function returns the projection operator onto \f$S\f$ for
105 : * \f$d+1\f$ dimensional quantities:
106 : *
107 : * \f{align*}
108 : * P_{ab} = g_{ab} + n_a n_b - s_a s_b = \gamma_{ab} - s_a s_b.
109 : * \f}
110 : *
111 : * Here \f$n_a = g_{ab} n^b\f$ and \f$s_a = g_{ab} s^b\f$ are the spacetime
112 : * one-forms corresponding to the spacetime vectors \f$n^a\f$ and \f$s^a\f$.
113 : *
114 : * If \f$t^a=(1,0,0,0)\f$ is a vector in the time direction, then the unit
115 : * normal to the spatial slice \f$n^a\f$ is determined by the relation
116 : * \f$t^a = \alpha n^a + \beta^a\f$ (e.g. Eq. (2.98) of \cite BaumgarteShapiro),
117 : * where \f$\alpha\f$ is the lapse, \f$\beta^a = (0, \beta^i)\f$, and
118 : * \f$\beta^i\f$ is the shift. Solving for \f$n^a\f$ then gives
119 : * \f$n^a = \alpha^{-1}(t^a - \beta^a)\f$. Then since \f$n_a = g_{ab} n^b\f$,
120 : * the normal one-form is given by
121 : * \f$n_a = g_{ab} n^b = \alpha^{-1}(g_{at} - g_{ab} \beta^b)\f$. This implies
122 : * \f$n_i = \alpha^{-1}(g_{it} - g_{ij} \beta^j)\f$, or
123 : * \f$ n_i = -\alpha^{-1}(\beta_i - \beta_i) = 0\f$. Only \f$n_t\f$ is nonzero:
124 : * it is \f$n_t = \alpha^{-1}(g_{tt} - g_{tj} \beta^j)\f$, or
125 : * \f$n_t = \alpha^{-1}(-\alpha^2 + \beta_j \beta^j - \beta_j \beta^j)\f$, or
126 : * \f$n_t = -\alpha\f$. Note that \f$n^a\f$ is a unit timelike vector, since
127 : * \f$n^a n_a = n^t n_t = \alpha^{-1}(-\alpha) = -1\f$.
128 : *
129 : * The unit normal to the boundary \f$s^a\f$ is orthogonal to \f$n^a\f$ and has
130 : * components \f$s^a = (0, s^i)\f$, where \f$s^i\f$ is the spatial unit normal
131 : * vector to the boundary. Since it is a spacelike unit normal vector whose
132 : * time component vanishes, \f$s^a s_a = s^i s_i = 1\f$.
133 : * Note that \f$s_a = g_{ab} s^b = g_{aj} s^j\f$, so
134 : * \f$s_i = g_{ij} s^j = \gamma_{ij} s^j\f$, where \f$\gamma_{ij}\f$ is the
135 : * spatial metric, while \f$s_t = g_{tj} s^j = \beta_j s^j = \beta^j s_j\f$.
136 : * Thus \f$n^a\f$ and \f$s_a\f$ are orthogonal, since
137 : * \f$n^a s_a = \alpha^{-1}(t^a s_a - \beta^a s_a)\f$, or
138 : * \f$n^a s_a = \alpha^{-1}(\beta^j s_j - \beta^j s_j) = 0\f$.
139 : *
140 : * This function computes $s_a$ from the inputs $s_i$ (provided as
141 : * `interface_unit_normal_one_form`) and \f$\beta^i\f$ (provided as `shift`).
142 : */
143 : template <typename DataType, size_t VolumeDim, typename Frame>
144 1 : tnsr::aa<DataType, VolumeDim, Frame> transverse_projection_operator(
145 : const tnsr::aa<DataType, VolumeDim, Frame>& spacetime_metric,
146 : const tnsr::a<DataType, VolumeDim, Frame>& spacetime_normal_one_form,
147 : const tnsr::i<DataType, VolumeDim, Frame>& interface_unit_normal_one_form,
148 : const tnsr::I<DataType, VolumeDim, Frame>& shift);
149 :
150 : template <typename DataType, size_t VolumeDim, typename Frame>
151 1 : void transverse_projection_operator(
152 : gsl::not_null<tnsr::aa<DataType, VolumeDim, Frame>*> projection_tensor,
153 : const tnsr::aa<DataType, VolumeDim, Frame>& spacetime_metric,
154 : const tnsr::a<DataType, VolumeDim, Frame>& spacetime_normal_one_form,
155 : const tnsr::i<DataType, VolumeDim, Frame>& interface_unit_normal_one_form,
156 : const tnsr::I<DataType, VolumeDim, Frame>& shift);
157 : /// @}
158 :
159 : /*!
160 : * \ingroup GeneralRelativityGroup
161 : * \brief Compute spacetime projection operator onto an interface
162 : *
163 : * \details Consider a \f$d-1\f$-dimensional surface \f$S\f$ in a
164 : * \f$d\f$-dimensional spatial hypersurface \f$\Sigma\f$. Let \f$s^a\f$
165 : * be the unit spacelike vector orthogonal to \f$S\f$ in \f$\Sigma\f$,
166 : * and \f$n^a\f$ be the timelike unit vector orthogonal to \f$\Sigma\f$.
167 : * This function returns the projection operator onto \f$S\f$ for
168 : * \f$d+1\f$ dimensional quantities:
169 : *
170 : * \f{align*}
171 : * P^{ab} = g^{ab} + n^a n^b - s^a s^b = \gamma_{ab} - s_a s_b.
172 : * \f}
173 : */
174 : template <typename DataType, size_t VolumeDim, typename Frame>
175 1 : tnsr::AA<DataType, VolumeDim, Frame> transverse_projection_operator(
176 : const tnsr::AA<DataType, VolumeDim, Frame>& inverse_spacetime_metric,
177 : const tnsr::A<DataType, VolumeDim, Frame>& spacetime_normal_vector,
178 : const tnsr::I<DataType, VolumeDim, Frame>& interface_unit_normal_vector);
179 :
180 : /*!
181 : * \ingroup GeneralRelativityGroup
182 : * \brief Compute spacetime projection operator onto an interface
183 : *
184 : * \details Consider a \f$d-1\f$-dimensional surface \f$S\f$ in a
185 : * \f$d\f$-dimensional spatial hypersurface \f$\Sigma\f$. Let \f$s^a\f$
186 : * be the unit spacelike vector orthogonal to \f$S\f$ in \f$\Sigma\f$,
187 : * and \f$n^a\f$ be the timelike unit vector orthogonal to \f$\Sigma\f$.
188 : * This function returns the projection operator onto \f$S\f$ for
189 : * \f$d+1\f$ dimensional quantities:
190 : *
191 : * \f{align*}
192 : * P^{ab} = g^{ab} + n^a n^b - s^a s^b = \gamma_{ab} - s_a s_b.
193 : * \f}
194 : */
195 : template <typename DataType, size_t VolumeDim, typename Frame>
196 1 : void transverse_projection_operator(
197 : gsl::not_null<tnsr::AA<DataType, VolumeDim, Frame>*> projection_tensor,
198 : const tnsr::AA<DataType, VolumeDim, Frame>& inverse_spacetime_metric,
199 : const tnsr::A<DataType, VolumeDim, Frame>& spacetime_normal_vector,
200 : const tnsr::I<DataType, VolumeDim, Frame>& interface_unit_normal_vector);
201 :
202 : /// @{
203 : /*!
204 : * \ingroup GeneralRelativityGroup
205 : * \brief Compute spacetime projection operator onto an interface
206 : *
207 : * \details Consider a \f$d-1\f$-dimensional surface \f$S\f$ in a
208 : * \f$d\f$-dimensional spatial hypersurface \f$\Sigma\f$. Let \f$s^a\f$
209 : * \f$(s_a)\f$ be the unit spacelike vector (one-form) orthogonal
210 : * to \f$S\f$ in \f$\Sigma\f$, and \f$n^a\f$ \f$(n_a)\f$ be the timelike
211 : * unit vector (one-form) orthogonal to \f$\Sigma\f$. This function
212 : * returns the projection operator onto \f$S\f$ for \f$d+1\f$ dimensional
213 : * quantities:
214 : *
215 : * \f{align*}
216 : * P^a_b = \delta^a_b + n^a n_b - s^a s_b.
217 : * \f}
218 : *
219 : * Here \f$n_a = g_{ab} n^b\f$ and \f$s_a = g_{ab} s^b\f$ are the spacetime
220 : * one-forms corresponding to the spacetime vectors \f$n^a\f$ and \f$s^a\f$.
221 : *
222 : * If \f$t^a=(1,0,0,0)\f$ is a vector in the time direction, then the unit
223 : * normal to the spatial slice \f$n^a\f$ is determined by the relation
224 : * \f$t^a = \alpha n^a + \beta^a\f$ (e.g. Eq. (2.98) of \cite BaumgarteShapiro),
225 : * where \f$\alpha\f$ is the lapse, \f$\beta^a = (0, \beta^i)\f$, and
226 : * \f$\beta^i\f$ is the shift. Solving for \f$n^a\f$ then gives
227 : * \f$n^a = \alpha^{-1}(t^a - \beta^a)\f$. Then since \f$n_a = g_{ab} n^b\f$,
228 : * the normal one-form is given by
229 : * \f$n_a = g_{ab} n^b = \alpha^{-1}(g_{at} - g_{ab} \beta^b)\f$. This implies
230 : * \f$n_i = \alpha^{-1}(g_{it} - g_{ij} \beta^j)\f$, or
231 : * \f$ n_i = -\alpha^{-1}(\beta_i - \beta_i) = 0\f$. Only \f$n_t\f$ is nonzero:
232 : * it is \f$n_t = \alpha^{-1}(g_{tt} - g_{tj} \beta^j)\f$, or
233 : * \f$n_t = \alpha^{-1}(-\alpha^2 + \beta_j \beta^j - \beta_j \beta^j)\f$, or
234 : * \f$n_t = -\alpha\f$. Note that \f$n^a\f$ is a unit timelike vector, since
235 : * \f$n^a n_a = n^t n_t = \alpha^{-1}(-\alpha) = -1\f$.
236 : *
237 : * The unit normal to the boundary \f$s^a\f$ is orthogonal to \f$n^a\f$ and has
238 : * components \f$s^a = (0, s^i)\f$, where \f$s^i\f$ is the spatial unit normal
239 : * vector to the boundary. Since it is a spacelike unit normal vector whose
240 : * time component vanishes, \f$s^a s_a = s^i s_i = 1\f$.
241 : * Note that \f$s_a = g_{ab} s^b = g_{aj} s^j\f$, so
242 : * \f$s_i = g_{ij} s^j = \gamma_{ij} s^j\f$, where \f$\gamma_{ij}\f$ is the
243 : * spatial metric, while \f$s_t = g_{tj} s^j = \beta_j s^j = \beta^j s_j\f$.
244 : * Thus \f$n^a\f$ and \f$s_a\f$ are orthogonal, since
245 : * \f$n^a s_a = \alpha^{-1}(t^a s_a - \beta^a s_a)\f$, or
246 : * \f$n^a s_a = \alpha^{-1}(\beta^j s_j - \beta^j s_j) = 0\f$.
247 : *
248 : * This function computes \f$s_a\f$ from the inputs \f$s_i\f$ (provided as
249 : * `interface_unit_normal_one_form`) and \f$\beta^i\f$ (provided as `shift`),
250 : * and it computes \f$s^a\f$ from \f$s^i\f$ (since \f$s^t=0\f$).
251 : */
252 : template <typename DataType, size_t VolumeDim, typename Frame>
253 1 : tnsr::Ab<DataType, VolumeDim, Frame> transverse_projection_operator(
254 : const tnsr::A<DataType, VolumeDim, Frame>& spacetime_normal_vector,
255 : const tnsr::a<DataType, VolumeDim, Frame>& spacetime_normal_one_form,
256 : const tnsr::I<DataType, VolumeDim, Frame>& interface_unit_normal_vector,
257 : const tnsr::i<DataType, VolumeDim, Frame>& interface_unit_normal_one_form,
258 : const tnsr::I<DataType, VolumeDim, Frame>& shift);
259 :
260 : template <typename DataType, size_t VolumeDim, typename Frame>
261 1 : void transverse_projection_operator(
262 : gsl::not_null<tnsr::Ab<DataType, VolumeDim, Frame>*> projection_tensor,
263 : const tnsr::A<DataType, VolumeDim, Frame>& spacetime_normal_vector,
264 : const tnsr::a<DataType, VolumeDim, Frame>& spacetime_normal_one_form,
265 : const tnsr::I<DataType, VolumeDim, Frame>& interface_unit_normal_vector,
266 : const tnsr::i<DataType, VolumeDim, Frame>& interface_unit_normal_one_form,
267 : const tnsr::I<DataType, VolumeDim, Frame>& shift);
268 : /// @}
269 : } // namespace gr
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