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 : * \ingroup GeneralRelativityGroup
21 : * \brief Compute null normal one-form to the boundary of a closed
22 : * region in a spatial slice of spacetime.
23 : *
24 : * \details Consider an \f$n-1\f$-dimensional boundary \f$S\f$ of a closed
25 : * region in an \f$n\f$-dimensional spatial hypersurface \f$\Sigma\f$. Let
26 : * \f$s^a\f$ be the unit spacelike vector orthogonal to \f$S\f$ in \f$\Sigma\f$,
27 : * and \f$n^a\f$ be the timelike unit vector orthogonal to \f$\Sigma\f$.
28 : * This function returns the null one-form that is outgoing/incoming on \f$S\f$:
29 : *
30 : * \f{align*}
31 : * k_a = \frac{1}{\sqrt{2}}\left(n_a \pm s_a\right).
32 : * \f}
33 : *
34 : * Here \f$n_a = g_{ab} n^b\f$ and \f$s_a = g_{ab} s^b\f$ are the spacetime
35 : * one-forms corresponding to the spacetime vectors \f$n^a\f$ and \f$s^a\f$.
36 : *
37 : * If \f$t^a=(1,0,0,0)\f$ is a vector in the time direction, then the unit
38 : * normal to the spatial slice \f$n^a\f$ is determined by the relation
39 : * \f$t^a = \alpha n^a + \beta^a\f$ (e.g. Eq. (2.98) of \cite BaumgarteShapiro),
40 : * where \f$\alpha\f$ is the lapse, \f$\beta^a = (0, \beta^i)\f$, and
41 : * \f$\beta^i\f$ is the shift. Solving for \f$n^a\f$ then gives
42 : * \f$n^a = \alpha^{-1}(t^a - \beta^a)\f$. Then since \f$n_a = g_{ab} n^b\f$,
43 : * the normal one-form is given by
44 : * \f$n_a = g_{ab} n^b = \alpha^{-1}(g_{at} - g_{ab} \beta^b)\f$. This implies
45 : * \f$n_i = \alpha^{-1}(g_{it} - g_{ij} \beta^j)\f$, or
46 : * \f$ n_i = -\alpha^{-1}(\beta_i - \beta_i) = 0\f$. Only \f$n_t\f$ is nonzero:
47 : * it is \f$n_t = \alpha^{-1}(g_{tt} - g_{tj} \beta^j)\f$, or
48 : * \f$n_t = \alpha^{-1}(-\alpha^2 + \beta_j \beta^j - \beta_j \beta^j)\f$, or
49 : * \f$n_t = -\alpha\f$. Note that \f$n^a\f$ is a unit timelike vector, since
50 : * \f$n^a n_a = n^t n_t = \alpha^{-1}(-\alpha) = -1\f$.
51 : *
52 : * The unit normal to the boundary \f$s^a\f$ is orthogonal to \f$n^a\f$ and has
53 : * components \f$s^a = (0, s^i)\f$, where \f$s^i\f$ is the spatial unit normal
54 : * vector to the boundary. Since it is a spacelike unit normal vector whose
55 : * time component vanishes, \f$s^a s_a = s^i s_i = 1\f$.
56 : * Note that \f$s_a = g_{ab} s^b = g_{aj} s^j\f$, so
57 : * \f$s_i = g_{ij} s^j = \gamma_{ij} s^j\f$, where \f$\gamma_{ij}\f$ is the
58 : * spatial metric, while \f$s_t = g_{tj} s^j = \beta_j s^j = \beta^j s_j\f$.
59 : * Thus \f$n^a\f$ and \f$s_a\f$ are orthogonal, since
60 : * \f$n^a s_a = \alpha^{-1}(t^a s_a - \beta^a s_a)\f$, or
61 : * \f$n^a s_a = \alpha^{-1}(\beta^j s_j - \beta^j s_j) = 0\f$.
62 : *
63 : * This function computes \f$s_a\f$ from the inputs \f$s_i\f$ (provided as
64 : * `interface_unit_normal_one_form`) and \f$\beta^i\f$ (provided as `shift`).
65 : */
66 : template <typename DataType, size_t VolumeDim, typename Frame>
67 1 : tnsr::a<DataType, VolumeDim, Frame> interface_null_normal(
68 : const tnsr::a<DataType, VolumeDim, Frame>& spacetime_normal_one_form,
69 : const tnsr::i<DataType, VolumeDim, Frame>& interface_unit_normal_one_form,
70 : const tnsr::I<DataType, VolumeDim, Frame>& shift, double sign);
71 :
72 : template <typename DataType, size_t VolumeDim, typename Frame>
73 1 : void interface_null_normal(
74 : gsl::not_null<tnsr::a<DataType, VolumeDim, Frame>*> null_one_form,
75 : const tnsr::a<DataType, VolumeDim, Frame>& spacetime_normal_one_form,
76 : const tnsr::i<DataType, VolumeDim, Frame>& interface_unit_normal_one_form,
77 : const tnsr::I<DataType, VolumeDim, Frame>& shift, double sign);
78 : /// @}
79 :
80 : /*!
81 : * \ingroup GeneralRelativityGroup
82 : * \brief Compute null normal vector to the boundary of a closed
83 : * region in a spatial slice of spacetime.
84 : *
85 : * \details Consider an \f$n-1\f$-dimensional boundary \f$S\f$ of a closed
86 : * region in an \f$n\f$-dimensional spatial hypersurface \f$\Sigma\f$. Let
87 : * \f$s^a\f$ be the unit spacelike vector orthogonal to \f$S\f$ in \f$\Sigma\f$,
88 : * and \f$n^a\f$ be the timelike unit vector orthogonal to \f$\Sigma\f$.
89 : * This function returns the null vector that is outgoing/ingoing on \f$S\f$:
90 : *
91 : * \f{align*}
92 : * k^a = \frac{1}{\sqrt{2}}\left(n^a \pm s^a\right).
93 : * \f}
94 : */
95 : template <typename DataType, size_t VolumeDim, typename Frame>
96 1 : tnsr::A<DataType, VolumeDim, Frame> interface_null_normal(
97 : const tnsr::A<DataType, VolumeDim, Frame>& spacetime_normal_vector,
98 : const tnsr::I<DataType, VolumeDim, Frame>& interface_unit_normal_vector,
99 : double sign);
100 :
101 : /*!
102 : * \ingroup GeneralRelativityGroup
103 : * \brief Compute null normal vector to the boundary of a closed
104 : * region in a spatial slice of spacetime.
105 : *
106 : * \details Consider an \f$n-1\f$-dimensional boundary \f$S\f$ of a closed
107 : * region in an \f$n\f$-dimensional spatial hypersurface \f$\Sigma\f$. Let
108 : * \f$s^a\f$ be the unit spacelike vector orthogonal to \f$S\f$ in \f$\Sigma\f$,
109 : * and \f$n^a\f$ be the timelike unit vector orthogonal to \f$\Sigma\f$.
110 : * This function returns the null vector that is outgoing/ingoing on \f$S\f$:
111 : *
112 : * \f{align*}
113 : * k^a = \frac{1}{\sqrt{2}}\left(n^a \pm s^a\right).
114 : * \f}
115 : */
116 : template <typename DataType, size_t VolumeDim, typename Frame>
117 1 : void interface_null_normal(
118 : gsl::not_null<tnsr::A<DataType, VolumeDim, Frame>*> null_vector,
119 : const tnsr::A<DataType, VolumeDim, Frame>& spacetime_normal_vector,
120 : const tnsr::I<DataType, VolumeDim, Frame>& interface_unit_normal_vector,
121 : double sign);
122 : } // namespace gr
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