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 gr {
      12              : /// @{
      13              : /*!
      14              :  * \ingroup GeneralRelativityGroup
      15              :  * \brief Computes spacetime derivative of
      16              :  * \f$ \mathfrak{g}^{ab}\equiv (-g)^{1/2} g^{ab} \f$.
      17              :  *
      18              :  * \details Computes the spacetime derivative of
      19              :  * \f$ \mathfrak{g}^{ab}\equiv (-g)^{1/2} g^{ab} \f$, defined
      20              :  * in \cite Misner1973. Using \f$ (-g)^{1/2} = \alpha (\gamma)^{1/2} \f$, where
      21              :  * \f$ \alpha \f$ is the lapse and \f$ \gamma \f$ is the determinant of the
      22              :  * spatial metric (\cite BaumgarteShapiro), the derivative of
      23              :  * \f$ \mathfrak{g}^{ab} \f$ expands out to
      24              :  * \f{align}{
      25              :  *   \partial_c \mathfrak{g}^{ab} &= \left[\partial_c \alpha \gamma^{1/2}
      26              :  *   + \frac{1}{2} \alpha \partial_c \gamma \gamma^{-1/2} \right] g^{ab}
      27              :  *   - \alpha \gamma^{1/2} g^{ad} g^{be} \partial_c g_{de}.
      28              :  * \f}
      29              :  */
      30              : template <typename DataType, size_t SpatialDim, typename Frame>
      31            1 : void spacetime_deriv_of_goth_g(
      32              :     gsl::not_null<tnsr::aBB<DataType, SpatialDim, Frame>*> da_goth_g,
      33              :     const tnsr::AA<DataType, SpatialDim, Frame>& inverse_spacetime_metric,
      34              :     const tnsr::abb<DataType, SpatialDim, Frame>& da_spacetime_metric,
      35              :     const Scalar<DataType>& lapse,
      36              :     const tnsr::a<DataType, SpatialDim, Frame>& da_lapse,
      37              :     const Scalar<DataType>& sqrt_det_spatial_metric,
      38              :     const tnsr::a<DataType, SpatialDim, Frame>& da_det_spatial_metric);
      39              : 
      40              : template <typename DataType, size_t SpatialDim, typename Frame>
      41            1 : tnsr::aBB<DataType, SpatialDim, Frame> spacetime_deriv_of_goth_g(
      42              :     const tnsr::AA<DataType, SpatialDim, Frame>& inverse_spacetime_metric,
      43              :     const tnsr::abb<DataType, SpatialDim, Frame>& da_spacetime_metric,
      44              :     const Scalar<DataType>& lapse,
      45              :     const tnsr::a<DataType, SpatialDim, Frame>& da_lapse,
      46              :     const Scalar<DataType>& sqrt_det_spatial_metric,
      47              :     const tnsr::a<DataType, SpatialDim, Frame>& da_det_spatial_metric);
      48              : /// @}
      49              : }  // namespace gr