Line data Source code
1 1 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : ///\file
5 : /// Declares function templates to calculate the Ricci tensor
6 :
7 : #pragma once
8 :
9 : #include <cstddef>
10 :
11 : #include "DataStructures/DataBox/Tag.hpp"
12 : #include "DataStructures/Tensor/EagerMath/RaiseOrLowerIndex.hpp"
13 : #include "DataStructures/Tensor/TypeAliases.hpp"
14 :
15 : /// \cond
16 : namespace gsl {
17 : template <typename>
18 : struct not_null;
19 : } // namespace gsl
20 : /// \endcond
21 :
22 : namespace gh {
23 : /// @{
24 : /*!
25 : * \ingroup GeneralRelativityGroup
26 : * \brief Compute spatial Ricci tensor using evolved variables and
27 : * their first derivatives.
28 : *
29 : * \details Lets write the Christoffel symbols of the first kind as
30 : * \f{align}
31 : * \Gamma_{kij} = \frac{1}{2}(\partial_i \gamma_{jk} +
32 : * \partial_j \gamma_{ik} -
33 : * \partial_k \gamma_{ij})
34 : * = (\Phi_{(ij)k} - \frac{1}{2}\Phi_{kij})
35 : * \f}
36 : * substituting \f$\partial_k \gamma_{ij}\rightarrow{}\Phi_{kij}\f$ by
37 : * subtracting out the three-index constraint
38 : * \f$C_{kij}=\partial_{k}\gamma_{ij}-\Phi_{kij}\f$ from every term. We also
39 : * define contractions \f$d_k=\frac{1}{2}\gamma^{ij}\Phi_{kij}\f$ and
40 : * \f$b_k=\frac{1}{2}\gamma^{ij}\Phi_{ijk}\f$. This allows us to rewrite the
41 : * spatial Ricci tensor as:
42 : * \f{align}
43 : * R_{i j} =& \partial_k \Gamma^{k}_{ij} - \partial_i \Gamma^{k}_{kj}
44 : * + \Gamma^{k}_{kl}\Gamma^{l}_{ij}
45 : * - \Gamma^{l}_{ki}\Gamma^{k}_{lj},\\
46 : *
47 : * =& \gamma^{kl}\left(\partial_{k}\Phi_{(ij)l} -
48 : * \frac{1}{2}\partial_{k}\Phi_{lij}\right)
49 : * - b^{l} (\Phi_{(ij)l} - \frac{1}{2}\Phi_{lij})\nonumber\\
50 : * & - \gamma^{kl}\left(\partial_{i}\Phi_{(kj)l}
51 : * - \frac{1}{2}\partial_{i}\Phi_{lkj}\right)
52 : * - \Phi_{i}{}^{kl}\left(\Phi_{(kj)l}
53 : * - \frac{1}{2}\Phi_{lkj}
54 : * \right)\nonumber\\
55 : * & + \gamma^{km}\left(\Phi_{(kl)m} - \frac{1}{2}\Phi_{mkl}\right)
56 : * \gamma^{ln}\left(\Phi_{(ij)n} - \frac{1}{2}\Phi_{nij}\right)
57 : * \nonumber\\
58 : *
59 : * & - \gamma^{km}\left(\Phi_{(il)m} - \frac{1}{2}\Phi_{mil}\right)
60 : * \gamma^{ln}\left(\Phi_{(jk)n} - \frac{1}{2}\Phi_{njk}\right).
61 : * \f}
62 : * Gathering all terms with second derivatives:
63 : * \f{align}
64 : * R_{i j} =& \frac{1}{2} \gamma^{k l} \left(\partial_k\Phi_{ijl}
65 : * + \partial_k\Phi_{jil}
66 : * - \partial_k\Phi_{lij}
67 : * + \partial_i\Phi_{lkj}
68 : * - \partial_i\Phi_{kjl}
69 : * - \partial_i\Phi_{jkl}\right)
70 : * + \mathcal{O}(\Phi), \nonumber\\
71 : * =& \frac{1}{2} \gamma^{kl} \left(\partial_{(j}\Phi_{lki)}
72 : * - \partial_{(j}\Phi_{i)kl}
73 : * + \partial_k \Phi_{(ij)l}
74 : * - \partial_l \Phi_{kij} \right)
75 : * + \mathcal{O}(\Phi),
76 : * \f}
77 : * where we use the four-index constraint
78 : * \f$C_{klij}=\partial_k\Phi_{lij}-\partial_l\Phi_{kij}=0\f$ to swap the
79 : * first and second derivatives of the spatial metric, and symmetrize
80 : * \f$R_{ij} = R_{(ij)}\f$. Similarly gathering the remaining terms and
81 : * using the four-index constraint we get:
82 : * \f{align}
83 : * R_{i j} =& - b^k\left(\Phi_{ijk} + \Phi_{jik} - \Phi_{kij}\right)
84 : * -\frac{1}{2} \Phi_i{}^{kl} \left(\Phi_{jkl} + \Phi_{kjl}
85 : * - \Phi_{lkj}\right)\nonumber\\
86 : *
87 : * &+ \frac{1}{2} d^k \left(\Phi_{ijk} + \Phi_{jik} - \Phi_{kij}\right)
88 : * - \left(\Phi_{(il)}{}^k - \frac{1}{2} \Phi^k{}_{il}\right)
89 : * \left(\Phi_{(kj)}{}^l - \frac{1}{2} \Phi^l{}_{kj}\right)
90 : * + \mathcal{O}(\partial\Phi) \\
91 : *
92 : * =& \frac{1}{2} \left(\Phi_{ijk} + \Phi_{jik} - \Phi_{kij}\right)
93 : * (d^k - 2 b^k)
94 : * + \frac{1}{4} \Phi_{ik}{}^l \Phi_{jl}{}^k
95 : * + \frac{1}{2} \left(\Phi^k{}_{il} \Phi_{kj}{}^l
96 : * - \Phi^k{}_{li} \Phi^l{}_{kj}\right)
97 : * + \mathcal{O}(\partial\Phi).
98 : * \f}
99 : * Gathering everything together, we compute the spatial Ricci tensor as:
100 : * \f{eqnarray}
101 : * R_{i j} &=& \frac{1}{2} \gamma^{kl} \left(\partial_{(j|}\Phi_{lk|i)}
102 : * - \partial_{(j}\Phi_{i)kl}
103 : * + \partial_k \Phi_{(ij)l}
104 : * - \partial_l \Phi_{kij}\right)\nonumber\\
105 : * &+& \frac{1}{2} \left(\Phi_{ijk} + \Phi_{jik} - \Phi_{kij}\right)
106 : * (d^k - 2 b^k)
107 : * + \frac{1}{4} \Phi_{ik}{}^l \Phi_{jl}{}^k
108 : * + \frac{1}{2} \left(\Phi^k{}_{il} \Phi_{kj}{}^l
109 : * - \Phi^k{}_{li} \Phi^l{}_{kj}\right).
110 : * \label{eq:rij}
111 : * \f}
112 : * This follows from equations (2.13) - (2.20) of \cite Kidder2001tz .
113 : *
114 : * Note that, in code, the mixed-index variables \f$\Phi_{ij}{}^k\f$ and
115 : * \f$\Phi^i{}_{jk}\f$ in Eq.(\f$\ref{eq:rij}\f$) are computed with a factor of
116 : * \f$1/2\f$ and so the last 3 terms in the same equation that are quadratic in
117 : * these terms occur multiplied by a factor of \f$4\f$.
118 : */
119 : template <typename DataType, size_t VolumeDim, typename Frame>
120 1 : void spatial_ricci_tensor(
121 : gsl::not_null<tnsr::ii<DataType, VolumeDim, Frame>*> ricci,
122 : const tnsr::iaa<DataType, VolumeDim, Frame>& phi,
123 : const tnsr::ijaa<DataType, VolumeDim, Frame>& deriv_phi,
124 : const tnsr::II<DataType, VolumeDim, Frame>& inverse_spatial_metric);
125 :
126 : template <typename DataType, size_t VolumeDim, typename Frame>
127 1 : tnsr::ii<DataType, VolumeDim, Frame> spatial_ricci_tensor(
128 : const tnsr::iaa<DataType, VolumeDim, Frame>& phi,
129 : const tnsr::ijaa<DataType, VolumeDim, Frame>& deriv_phi,
130 : const tnsr::II<DataType, VolumeDim, Frame>& inverse_spatial_metric);
131 : /// @}
132 : } // namespace gh
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