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