BSSNAtildeTensor.hpp

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#pragma once
 
#include "../../Core/Derivate.hpp"
#include "../../Core/Matrix.hpp"
#include "../../Core/Tensor.hpp"
#include "../../Core/Vector.hpp"
#include "../Metric.hpp"
#include "BSSNSetup.hpp"
 
namespace tensorium_RG {
template <typename K> class BSSNAtildeTensor {
  public:
    using Vec = tensorium::Vector<K>;
    using Mat = tensorium::Tensor<K, 2>;
    tensorium::Tensor<K, 2> compute_Atilde_tensor(const tensorium::Tensor<K, 2> &Kij,
                                                  const tensorium::Tensor<K, 2> &gamma_inv,
                                                  const tensorium::Tensor<K, 2> &gamma, K chi) {
        K trace_K = K(0);
        for (size_t i = 0; i < 3; ++i)
            for (size_t j = 0; j < 3; ++j)
                trace_K += gamma_inv(i, j) * Kij(i, j);
 
        tensorium::Tensor<K, 2> Atilde({3, 3});
        for (size_t i = 0; i < 3; ++i)
            for (size_t j = 0; j < 3; ++j)
                Atilde(i, j) = chi * (Kij(i, j) - (1. / 3.) * gamma(i, j) * trace_K);
 
        // Ensure symmetry: \f$ \tilde{A}_{ij} = \tilde{A}_{ji} \f$
        for (size_t i = 0; i < 3; ++i)
            for (size_t j = i + 1; j < 3; ++j) {
                K sym = (Atilde(i, j) + Atilde(j, i)) * 0.5;
                Atilde(i, j) = Atilde(j, i) = sym;
            }
 
        return Atilde;
    }
};
 
} // namespace tensorium_RG