Line data Source code
1 0 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : #pragma once
5 :
6 : #include <string>
7 :
8 : #include "DataStructures/DataBox/Tag.hpp"
9 : #include "DataStructures/Tensor/TypeAliases.hpp"
10 : #include "PointwiseFunctions/GeneralRelativity/Tags.hpp"
11 : #include "PointwiseFunctions/GeneralRelativity/Tags/Conformal.hpp"
12 :
13 : namespace Xcts {
14 : /// Tags related to the XCTS equations
15 : namespace Tags {
16 :
17 : /*!
18 : * \brief The conformal factor \f$\psi(x)\f$ that rescales the spatial metric
19 : * \f$\gamma_{ij}=\psi^4\bar{\gamma}_{ij}\f$.
20 : */
21 : template <typename DataType>
22 1 : struct ConformalFactor : db::SimpleTag {
23 0 : using type = Scalar<DataType>;
24 : };
25 :
26 : /*!
27 : * \brief The conformal factor minus one \f$\psi(x) - 1\f$. Useful as dynamic
28 : * variable in formulations of the XCTS equations because it approaches zero at
29 : * spatial infinity rather than one, hence derivatives may be more accurate.
30 : */
31 : template <typename DataType>
32 1 : struct ConformalFactorMinusOne : db::SimpleTag {
33 0 : using type = Scalar<DataType>;
34 : };
35 :
36 : /*!
37 : * \brief The conformally scaled spatial metric
38 : * \f$\bar{\gamma}_{ij}=\psi^{-4}\gamma_{ij}\f$, where \f$\psi\f$ is the
39 : * `Xcts::Tags::ConformalFactor` and \f$\gamma_{ij}\f$ is the
40 : * `gr::Tags::SpatialMetric`
41 : */
42 : template <typename DataType, size_t Dim, typename Frame>
43 1 : using ConformalMetric =
44 : gr::Tags::Conformal<gr::Tags::SpatialMetric<DataType, Dim, Frame>, -4>;
45 :
46 : /*!
47 : * \brief The conformally scaled inverse spatial metric
48 : * \f$\bar{\gamma}^{ij}=\psi^{4}\gamma^{ij}\f$, where \f$\psi\f$ is the
49 : * `Xcts::Tags::ConformalFactor` and \f$\gamma^{ij}\f$ is the
50 : * `gr::Tags::InverseSpatialMetric`
51 : */
52 : template <typename DataType, size_t Dim, typename Frame>
53 1 : using InverseConformalMetric =
54 : gr::Tags::Conformal<gr::Tags::InverseSpatialMetric<DataType, Dim, Frame>,
55 : 4>;
56 :
57 : /*!
58 : * \brief The product of lapse \f$\alpha(x)\f$ and conformal factor
59 : * \f$\psi(x)\f$
60 : *
61 : * This quantity is commonly used in formulations of the XCTS equations.
62 : */
63 : template <typename DataType>
64 1 : struct LapseTimesConformalFactor : db::SimpleTag {
65 0 : using type = Scalar<DataType>;
66 : };
67 :
68 : /*!
69 : * \brief The lapse times the conformal factor minus one \f$\alpha \psi - 1\f$.
70 : *
71 : * \see `Xcts::Tags::ConformalFactorMinusOne`
72 : */
73 : template <typename DataType>
74 1 : struct LapseTimesConformalFactorMinusOne : db::SimpleTag {
75 0 : using type = Scalar<DataType>;
76 : };
77 :
78 : /*!
79 : * \brief The constant part \f$\beta^i_\mathrm{background}\f$ of the shift
80 : * \f$\beta^i=\beta^i_\mathrm{background} + \beta^i_\mathrm{excess}\f$
81 : *
82 : * \see `Xcts::Tags::ShiftExcess`
83 : */
84 : template <typename DataType, size_t Dim, typename Frame>
85 1 : struct ShiftBackground : db::SimpleTag {
86 0 : using type = tnsr::I<DataType, Dim, Frame>;
87 : };
88 :
89 : /*!
90 : * \brief The dynamic part \f$\beta^i_\mathrm{excess}\f$ of the shift
91 : * \f$\beta^i=\beta^i_\mathrm{background} + \beta^i_\mathrm{excess}\f$
92 : *
93 : * We commonly split off the part of the shift that diverges at large coordinate
94 : * distances (the "background" shift \f$\beta^i_\mathrm{background}\f$) and
95 : * solve only for the remainder (the "excess" shift
96 : * \f$\beta^i_\mathrm{excess}\f$). For example, the background shift might be a
97 : * uniform rotation \f$\beta^i_\mathrm{background}=(-\Omega y, \Omega x, 0)\f$
98 : * with angular velocity \f$\Omega\f$ around the z-axis, given here in Cartesian
99 : * coordinates.
100 : *
101 : * \see `Xcts::Tags::Background`
102 : */
103 : template <typename DataType, size_t Dim, typename Frame>
104 1 : struct ShiftExcess : db::SimpleTag {
105 0 : using type = tnsr::I<DataType, Dim, Frame>;
106 : };
107 :
108 : /*!
109 : * \brief The symmetric "strain" of the shift vector
110 : * \f$B_{ij} = \bar{D}_{(i}\bar{\gamma}_{j)k}\beta^k =
111 : * \left(\partial_{(i}\bar{\gamma}_{j)k} - \bar{\Gamma}_{kij}\right)\beta^k\f$
112 : *
113 : * This quantity is used in our formulations of the XCTS equations.
114 : *
115 : * Note that the shift is not a conformal quantity, so its index is generally
116 : * raised and lowered with the spatial metric, not with the conformal metric.
117 : * However, to compute this "strain" we use the conformal metric as defined
118 : * above. The conformal longitudinal shift in terms of this quantity is then:
119 : *
120 : * \f{equation}
121 : * (\bar{L}\beta)^{ij} = 2\left(\bar{\gamma}^{ik}\bar{\gamma}^{jl}
122 : * - \frac{1}{3}\bar{\gamma}^{ij}\bar{\gamma}^{kl}\right) B_{kl}
123 : * \f}
124 : *
125 : * Note that the conformal longitudinal shift is (minus) the "stress" quantity
126 : * of a linear elasticity system in which the shift takes the role of the
127 : * displacement vector and the definition of its "strain" remains the same. This
128 : * auxiliary elasticity system is formulated on an isotropic constitutive
129 : * relation based on the conformal metric with vanishing bulk modulus \f$K=0\f$
130 : * (not to be confused with the extrinsic curvature trace \f$K\f$ in this
131 : * context) and unit shear modulus \f$\mu=1\f$. See the
132 : * `Elasticity::FirstOrderSystem` and the
133 : * `Elasticity::ConstitutiveRelations::IsotropicHomogeneous` for details.
134 : */
135 : template <typename DataType, size_t Dim, typename Frame>
136 1 : struct ShiftStrain : db::SimpleTag {
137 0 : using type = tnsr::ii<DataType, Dim, Frame>;
138 : };
139 :
140 : /*!
141 : * \brief The conformal longitudinal operator applied to the shift excess
142 : * \f$(\bar{L}\beta_\mathrm{excess})^{ij}\f$
143 : *
144 : * This quantity can be used as the "flux" for the momentum constraint in
145 : * formulations of the XCTS equations, because the principal part of the
146 : * momentum constraint is essentially the divergence of this quantity.
147 : */
148 : template <typename DataType, size_t Dim, typename Frame>
149 1 : struct LongitudinalShiftExcess : db::SimpleTag {
150 0 : using type = tnsr::II<DataType, Dim, Frame>;
151 : };
152 :
153 : /*!
154 : * \brief The conformal longitudinal operator applied to the background shift
155 : * vector minus the time derivative of the conformal metric
156 : * \f$(\bar{L}\beta_\mathrm{background})^{ij} - \bar{u}^{ij}\f$
157 : *
158 : * This quantity appears in formulation of the XCTS equations (see `Xcts`) and
159 : * serves to specify their free data \f$\bar{u}_{ij}\f$. It is combined with the
160 : * longitudinal background shift because the two quantities are degenerate.
161 : *
162 : * \note As usual for conformal quantities, the indices here are raised with the
163 : * conformal metric: \f$\bar{u}^{ij} = \bar{\gamma}^{ik}\bar{\gamma}^{jl}
164 : * \partial_t\bar{\gamma}_{kl}\f$.
165 : *
166 : * \see `Xcts::Tags::ShiftBackground`
167 : */
168 : template <typename DataType, size_t Dim, typename Frame>
169 1 : struct LongitudinalShiftBackgroundMinusDtConformalMetric : db::SimpleTag {
170 0 : using type = tnsr::II<DataType, Dim, Frame>;
171 : };
172 :
173 : /*!
174 : * \brief The conformal longitudinal operator applied to the shift vector minus
175 : * the time derivative of the conformal metric, squared:
176 : * \f$\left((\bar{L}\beta)^{ij} - \bar{u}^{ij}\right)
177 : * \left((\bar{L}\beta)_{ij} - \bar{u}_{ij}\right)\f$
178 : */
179 : template <typename DataType>
180 1 : struct LongitudinalShiftMinusDtConformalMetricSquare : db::SimpleTag {
181 0 : using type = Scalar<DataType>;
182 : };
183 :
184 : /*!
185 : * \brief The conformal longitudinal operator applied to the shift vector minus
186 : * the time derivative of the conformal metric, squared and divided by the
187 : * square of the lapse:
188 : * \f$\frac{1}{\alpha^2}\left((\bar{L}\beta)^{ij} - \bar{u}^{ij}\right)
189 : * \left((\bar{L}\beta)_{ij} - \bar{u}_{ij}\right)\f$
190 : */
191 : template <typename DataType>
192 1 : struct LongitudinalShiftMinusDtConformalMetricOverLapseSquare : db::SimpleTag {
193 0 : using type = Scalar<DataType>;
194 : };
195 :
196 : /*!
197 : * \brief The shift vector contracted with the gradient of the trace of the
198 : * extrinsic curvature: \f$\beta^i\partial_i K\f$
199 : *
200 : * The shift vector in this quantity is the full shift
201 : * \f$\beta^i=\beta^i_\mathrm{background}+\beta^i_\mathrm{excess}\f$ (see
202 : * `Xcts::Tags::ShiftExcess` for details on this split).
203 : */
204 : template <typename DataType>
205 1 : struct ShiftDotDerivExtrinsicCurvatureTrace : db::SimpleTag {
206 0 : using type = Scalar<DataType>;
207 : };
208 :
209 : /*!
210 : * \brief The Christoffel symbols of the first kind related to the conformal
211 : * metric \f$\bar{\gamma}_{ij}\f$
212 : */
213 : template <typename DataType, size_t Dim, typename Frame>
214 1 : struct ConformalChristoffelFirstKind : db::SimpleTag {
215 0 : using type = tnsr::ijj<DataType, Dim, Frame>;
216 : };
217 :
218 : /*!
219 : * \brief The Christoffel symbols of the second kind related to the conformal
220 : * metric \f$\bar{\gamma}_{ij}\f$
221 : */
222 : template <typename DataType, size_t Dim, typename Frame>
223 1 : struct ConformalChristoffelSecondKind : db::SimpleTag {
224 0 : using type = tnsr::Ijj<DataType, Dim, Frame>;
225 : };
226 :
227 : /*!
228 : * \brief The Christoffel symbols of the second kind (related to the conformal
229 : * metric \f$\bar{\gamma}_{ij}\f$) contracted in their first two indices:
230 : * \f$\bar{\Gamma}_k = \bar{\Gamma}^{i}_{ik}\f$
231 : */
232 : template <typename DataType, size_t Dim, typename Frame>
233 1 : struct ConformalChristoffelContracted : db::SimpleTag {
234 0 : using type = tnsr::i<DataType, Dim, Frame>;
235 : };
236 :
237 : /*!
238 : * \brief The Ricci tensor related to the conformal metric
239 : * \f$\bar{\gamma}_{ij}\f$
240 : */
241 : template <typename DataType, size_t Dim, typename Frame>
242 1 : struct ConformalRicciTensor : db::SimpleTag {
243 0 : using type = tnsr::ii<DataType, Dim, Frame>;
244 : };
245 :
246 : /*!
247 : * \brief The Ricci scalar related to the conformal metric
248 : * \f$\bar{\gamma}_{ij}\f$
249 : */
250 : template <typename DataType>
251 1 : struct ConformalRicciScalar : db::SimpleTag {
252 0 : using type = Scalar<DataType>;
253 : };
254 :
255 : } // namespace Tags
256 : } // namespace Xcts
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