Ricci.hpp

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// Distributed under the MIT License.
// See LICENSE.txt for details.

///\file
/// Declares function templates to calculate the Ricci tensor

#pragma once

#include <cstddef>

#include "DataStructures/Tensor/TypeAliases.hpp"

namespace gr {

/*!
 * \ingroup GeneralRelativityGroup
 * \brief Computes Ricci tensor from the (spatial or spacetime)
 * Christoffel symbol of the second kind and its derivative.
 *
 * \details Computes Ricci tensor \f$R_{ab}\f$ as:
 * \f$ R_{ab} = \frac{1}{2} ( \partial_c \Gamma^{c}_{ab} - \partial_{(b}
 * \Gamma^{c}_{a)c}
 * + \Gamma^{d}_{ab}\Gamma^{c}_{cd} - \Gamma^{d}_{ac} \Gamma^{c}_{bd}  ) \f$
 * where \f$\Gamma^{a}_{bc}\f$ is the Christoffel symbol of the second kind.
 */
template <size_t SpatialDim, typename Frame, IndexType Index, typename DataType>
tnsr::aa<DataType, SpatialDim, Frame, Index> ricci_tensor(
    const tnsr::Abb<DataType, SpatialDim, Frame, Index>& christoffel_2nd_kind,
    const tnsr::aBcc<DataType, SpatialDim, Frame, Index>&
        d_christoffel_2nd_kind) noexcept;

} // namespace gr