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