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             : #include "Utilities/Gsl.hpp"
      10             : 
      11             : namespace Ccz4 {
      12             : /// @{
      13             : /*!
      14             :  * \brief Computes the spatial Ricci tensor
      15             :  *
      16             :  * \details Computes the spatial Ricci tensor as:
      17             :  *
      18             :  * \f{align}
      19             :  *     R_{ij} &=
      20             :  *       \partial_m \Gamma^m_{ij} - \partial_j \Gamma^m_{im} +
      21             :  *       \Gamma^l_{ij} \Gamma^m_{lm} - \Gamma^l_{im} \Gamma^m_{lj}
      22             :  * \f}
      23             :  *
      24             :  * where
      25             :  *
      26             :  * \f{align}
      27             :  *     \partial_k \Gamma^m_{ij} &=
      28             :  *       \partial_k \tilde{\Gamma}^m_{ij} +
      29             :  *       2 D_k{}^{ml} (\tilde{\gamma}_{jl} P_i + \tilde{\gamma}_{il} P_j -
      30             :  *                     \tilde{\gamma}_{ij} P_l)\nonumber\\
      31             :  *       & - 2 \tilde{\gamma}^{ml} (D_{kjl} P_i + D_{kil} P_j - D_{kij} P_l) -
      32             :  *       \tilde{\gamma}^{ml} (
      33             :  *         \tilde{\gamma}_{jl} \partial_{(k} P_{i)} +
      34             :  *         \tilde{\gamma}_{il} \partial_{(k} P_{j)} -
      35             :  *         \tilde{\gamma}_{ij} \partial_{(k} P_{l)})
      36             :  * \f}
      37             :  *
      38             :  * \f$\Gamma^k_{ij}\f$ is the spatial christoffel symbols of the second kind
      39             :  * defined by `Ccz4::Tags::ChristoffelSecondKind`,
      40             :  * \f$\partial_m \tilde{\Gamma}^k_{ij}\f$ is the spatial derivative of the
      41             :  * conformal spatial christoffel symbols of the second kind defined by
      42             :  * `Ccz4::Tags::DerivConformalChristoffelSecondKind`, \f$\tilde{\gamma}_{ij}\f$
      43             :  * is the conformal spatial metric defined by `Ccz4::Tags::ConformalMetric`,
      44             :  * \f$\tilde{\gamma}^{ij}\f$ is the inverse conformal spatial metric defined by
      45             :  * `Ccz4::Tags::InverseConformalMetric`, \f$D_{ijk}\f$ is the CCZ4 auxiliary
      46             :  * variable defined by `Ccz4::Tags::FieldD`, \f$D_k{}^{ij}\f$ is the CCZ4
      47             :  * identity defined by `Ccz4::Tags::FieldDUp`, \f$P_i\f$ is the CCZ4 auxiliary
      48             :  * variable defined by `Ccz4::Tags::FieldP`, and \f$\partial_j P_{i}\f$ is its
      49             :  * spatial derivative.
      50             :  *
      51             :  * After substituting in the full expressions for \f$\partial_m \Gamma^m_{ij}\f$
      52             :  * and \f$\partial_j \Gamma^m_{im}\f$ and commuting terms with common
      53             :  * coefficients, the full equation becomes and is implemented as:
      54             :  *
      55             :  *  \f{align}{
      56             :  *     R_{ij} &=
      57             :  *       \partial_m \tilde{\Gamma}^m_{ij} -
      58             :  *       \partial_j \tilde{\Gamma}^m_{im}\nonumber\\
      59             :  *       & + 2 D_m{}^{ml} (\tilde{\gamma}_{jl} P_i + \tilde{\gamma}_{il} P_j -
      60             :  *                         \tilde{\gamma}_{ij} P_l) -
      61             :  *       2 \tilde{\gamma}^{ml} (
      62             :  *         D_{mjl} P_i + D_{mil} P_j - D_{mij} P_l)\nonumber\\
      63             :  *       & - 2 D_j{}^{ml} (\tilde{\gamma}_{ml} P_i + \tilde{\gamma}_{il} P_m -
      64             :  *                         \tilde{\gamma}_{im} P_l) +
      65             :  *       2 \tilde{\gamma}^{ml} (
      66             :  *         D_{jml} P_i + D_{jil} P_m - D_{jim} P_l)\nonumber\\
      67             :  *       & - \tilde{\gamma}^{ml} (
      68             :  *         \tilde{\gamma}_{jl} \partial_{(m} P_{i)} +
      69             :  *         \tilde{\gamma}_{il} \partial_{(m} P_{j)} -
      70             :  *         \tilde{\gamma}_{ij} \partial_{(m} P_{l)}) +
      71             :  *       \tilde{\gamma}^{ml} (
      72             :  *         \tilde{\gamma}_{ml} \partial_{(j} P_{i)} +
      73             :  *         \tilde{\gamma}_{il} \partial_{(j} P_{m)} -
      74             :  *         \tilde{\gamma}_{im} \partial_{(j} P_{l)})\nonumber\\
      75             :  *       & + \Gamma^l_{ij} \Gamma^m_{lm} - \Gamma^l_{im} \Gamma^m_{lj}
      76             :  * \f}
      77             :  *
      78             :  * The argument `contracted_christoffel_second_kind` corresponds to the
      79             :  * \f$\Gamma^m_{lm}\f$ term, the argument
      80             :  * `contracted_d_conformal_christoffel_difference` corresponds to the
      81             :  * \f$\partial_m \tilde{\Gamma}^m_{ij} - \partial_j \tilde{\Gamma}^m_{im}\f$
      82             :  * term, and the argument `contracted_field_d_up` corresponds to the
      83             :  * \f$D_m{}^{ml}\f$ term.
      84             :  */
      85             : template <typename DataType, size_t Dim, typename Frame>
      86           1 : void spatial_ricci_tensor(
      87             :     const gsl::not_null<tnsr::ii<DataType, Dim, Frame>*> result,
      88             :     const tnsr::Ijj<DataType, Dim, Frame>& christoffel_second_kind,
      89             :     const tnsr::i<DataType, Dim, Frame>& contracted_christoffel_second_kind,
      90             :     const tnsr::ij<DataType, Dim, Frame>&
      91             :         contracted_d_conformal_christoffel_difference,
      92             :     const tnsr::ii<DataType, Dim, Frame>& conformal_spatial_metric,
      93             :     const tnsr::II<DataType, Dim, Frame>& inverse_conformal_spatial_metric,
      94             :     const tnsr::ijj<DataType, Dim, Frame>& field_d,
      95             :     const tnsr::iJJ<DataType, Dim, Frame>& field_d_up,
      96             :     const tnsr::I<DataType, Dim, Frame>& contracted_field_d_up,
      97             :     const tnsr::i<DataType, Dim, Frame>& field_p,
      98             :     const tnsr::ij<DataType, Dim, Frame>& d_field_p);
      99             : 
     100             : template <typename DataType, size_t Dim, typename Frame>
     101           1 : tnsr::ii<DataType, Dim, Frame> spatial_ricci_tensor(
     102             :     const tnsr::Ijj<DataType, Dim, Frame>& christoffel_second_kind,
     103             :     const tnsr::i<DataType, Dim, Frame>& contracted_christoffel_second_kind,
     104             :     const tnsr::ij<DataType, Dim, Frame>&
     105             :         contracted_d_conformal_christoffel_difference,
     106             :     const tnsr::ii<DataType, Dim, Frame>& conformal_spatial_metric,
     107             :     const tnsr::II<DataType, Dim, Frame>& inverse_conformal_spatial_metric,
     108             :     const tnsr::ijj<DataType, Dim, Frame>& field_d,
     109             :     const tnsr::iJJ<DataType, Dim, Frame>& field_d_up,
     110             :     const tnsr::I<DataType, Dim, Frame>& contracted_field_d_up,
     111             :     const tnsr::i<DataType, Dim, Frame>& field_p,
     112             :     const tnsr::ij<DataType, Dim, Frame>& d_field_p);
     113             : /// @}
     114             : }  // namespace Ccz4