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Merge pull request #5865 from nikwit/kerr-schild-derivatives
Add derivatives of Kerr Schild quantities
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src/Evolution/Systems/CurvedScalarWave/Worldtube/KerrSchildDerivatives.cpp
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| // Distributed under the MIT License. | ||
| // See LICENSE.txt for details. | ||
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| #include "Evolution/Systems/CurvedScalarWave/Worldtube/KerrSchildDerivatives.hpp" | ||
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| #include <cstddef> | ||
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| #include "DataStructures/Tensor/EagerMath/DotProduct.hpp" | ||
| #include "DataStructures/Tensor/EagerMath/Magnitude.hpp" | ||
| #include "DataStructures/Tensor/EagerMath/Trace.hpp" | ||
| #include "DataStructures/Tensor/Tensor.hpp" | ||
| #include "Utilities/Gsl.hpp" | ||
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| namespace CurvedScalarWave::Worldtube { | ||
| tnsr::iAA<double, 3> spatial_derivative_inverse_ks_metric( | ||
| const tnsr::I<double, 3>& pos) { | ||
| const double r_sq = get(dot_product(pos, pos)); | ||
| const double r = sqrt(r_sq); | ||
| const double one_over_r = 1. / r; | ||
| const double one_over_r_2 = 1. / r_sq; | ||
| const double one_over_r_3 = one_over_r_2 * one_over_r; | ||
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| tnsr::iAA<double, 3> di_imetric{}; | ||
| tnsr::ii<double, 3> delta_ll{0.}; | ||
| tnsr::Ij<double, 3> delta_ul{0.}; | ||
| tnsr::i<double, 3> pos_lower{}; | ||
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| for (size_t i = 0; i < 3; ++i) { | ||
| delta_ll.get(i, i) = 1.; | ||
| delta_ul.get(i, i) = 1.; | ||
| pos_lower.get(i) = pos.get(i); | ||
| } | ||
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| const auto d_imetric_ij = tenex::evaluate<ti::i, ti::J, ti::K>( | ||
| one_over_r_3 * | ||
| (6. * pos(ti::J) * pos(ti::K) * pos_lower(ti::i) * one_over_r_2 - | ||
| 2. * delta_ul(ti::J, ti::i) * pos(ti::K) - | ||
| 2. * delta_ul(ti::K, ti::i) * pos(ti::J))); | ||
| const auto d_imetric_i0 = tenex::evaluate<ti::i, ti::J>( | ||
| one_over_r_2 * (-4. * pos_lower(ti::i) * pos(ti::J) * one_over_r_2 + | ||
| 2. * delta_ul(ti::J, ti::i))); | ||
| const auto d_imetric_00 = | ||
| tenex::evaluate<ti::i>(2. * pos_lower(ti::i) * one_over_r_3); | ||
| for (size_t i = 0; i < 3; ++i) { | ||
| di_imetric.get(i, 0, 0) = d_imetric_00.get(i); | ||
| for (size_t j = 0; j < 3; ++j) { | ||
| di_imetric.get(i, j + 1, 0) = d_imetric_i0.get(i, j); | ||
| for (size_t k = 0; k < 3; ++k) { | ||
| di_imetric.get(i, j + 1, k + 1) = d_imetric_ij.get(i, j, k); | ||
| } | ||
| } | ||
| } | ||
| return di_imetric; | ||
| } | ||
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| tnsr::iaa<double, 3> spatial_derivative_ks_metric( | ||
| const tnsr::aa<double, 3>& metric, | ||
| const tnsr::iAA<double, 3>& di_inverse_metric) { | ||
| tnsr::iaa<double, 3> di_metric{}; | ||
| tenex::evaluate<ti::i, ti::a, ti::b>( | ||
| make_not_null(&di_metric), -metric(ti::a, ti::c) * metric(ti::b, ti::d) * | ||
| di_inverse_metric(ti::i, ti::C, ti::D)); | ||
| return di_metric; | ||
| } | ||
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| tnsr::iiAA<double, 3> second_spatial_derivative_inverse_ks_metric( | ||
| const tnsr::I<double, 3>& pos) { | ||
| const double r_sq = get(dot_product(pos, pos)); | ||
| const double r = sqrt(r_sq); | ||
| const double one_over_r = 1. / r; | ||
| const double one_over_r_2 = 1. / r_sq; | ||
| const double one_over_r_3 = one_over_r_2 * one_over_r; | ||
| const double one_over_r_4 = one_over_r_2 * one_over_r_2; | ||
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| tnsr::iiAA<double, 3> dij_imetric{}; | ||
| tnsr::ii<double, 3> delta_ll{0.}; | ||
| tnsr::Ij<double, 3> delta_ul{0.}; | ||
| tnsr::i<double, 3> pos_lower{}; | ||
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| for (size_t i = 0; i < 3; ++i) { | ||
| delta_ll.get(i, i) = 1.; | ||
| delta_ul.get(i, i) = 1.; | ||
| pos_lower.get(i) = pos.get(i); | ||
| } | ||
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| const auto d2_imetric_ij = tenex::evaluate<ti::i, ti::j, ti::K, ti::L>( | ||
| one_over_r_3 * | ||
| (-2. * (delta_ul(ti::L, ti::i) * delta_ul(ti::K, ti::j) + | ||
| delta_ul(ti::K, ti::i) * delta_ul(ti::L, ti::j)) + | ||
| one_over_r_2 * | ||
| (6. * (delta_ll(ti::i, ti::j) * pos(ti::K) * pos(ti::L) + | ||
| delta_ul(ti::K, ti::i) * pos_lower(ti::j) * pos(ti::L) + | ||
| delta_ul(ti::K, ti::j) * pos_lower(ti::i) * pos(ti::L) + | ||
| delta_ul(ti::L, ti::i) * pos_lower(ti::j) * pos(ti::K) + | ||
| delta_ul(ti::L, ti::j) * pos_lower(ti::i) * pos(ti::K)) - | ||
| one_over_r_2 * 30. * pos_lower(ti::i) * pos_lower(ti::j) * | ||
| pos(ti::K) * pos(ti::L)))); | ||
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| const auto d2_imetric_i0 = tenex::evaluate<ti::j, ti::k, ti::I>( | ||
| one_over_r_4 * | ||
| (-4. * (delta_ll(ti::k, ti::j) * pos(ti::I) + | ||
| delta_ul(ti::I, ti::k) * pos_lower(ti::j) + | ||
| delta_ul(ti::I, ti::j) * pos_lower(ti::k)) + | ||
| one_over_r_2 * 16. * pos(ti::I) * pos_lower(ti::j) * pos_lower(ti::k))); | ||
| const auto d2_imetric_00 = tenex::evaluate<ti::i, ti::j>( | ||
| one_over_r_3 * (2. * delta_ll(ti::i, ti::j) - | ||
| one_over_r_2 * 6. * pos_lower(ti::i) * pos_lower(ti::j))); | ||
| for (size_t i = 0; i < 3; ++i) { | ||
| for (size_t j = 0; j < 3; ++j) { | ||
| dij_imetric.get(i, j, 0, 0) = d2_imetric_00.get(i, j); | ||
| for (size_t k = 0; k < 3; ++k) { | ||
| dij_imetric.get(i, j, k + 1, 0) = d2_imetric_i0.get(i, j, k); | ||
| for (size_t l = 0; l < 3; ++l) { | ||
| dij_imetric.get(i, j, k + 1, l + 1) = d2_imetric_ij.get(i, j, k, l); | ||
| } | ||
| } | ||
| } | ||
| } | ||
| return dij_imetric; | ||
| } | ||
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| tnsr::iiaa<double, 3> second_spatial_derivative_metric( | ||
| const tnsr::aa<double, 3>& metric, const tnsr::iaa<double, 3>& di_metric, | ||
| const tnsr::iAA<double, 3>& di_inverse_metric, | ||
| const tnsr::iiAA<double, 3>& dij_inverse_metric) { | ||
| tnsr::iiaa<double, 3> dij_metric{}; | ||
| tenex::evaluate<ti::j, ti::i, ti::a, ti::b>( | ||
| make_not_null(&dij_metric), | ||
| -metric(ti::a, ti::c) * metric(ti::b, ti::d) * | ||
| dij_inverse_metric(ti::j, ti::i, ti::C, ti::D) - | ||
| 2. * metric(ti::a, ti::c) * di_metric(ti::j, ti::b, ti::d) * | ||
| di_inverse_metric(ti::i, ti::C, ti::D)); | ||
| return dij_metric; | ||
| } | ||
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| tnsr::iAbb<double, 3> spatial_derivative_christoffel( | ||
| const tnsr::iaa<double, 3>& di_metric, | ||
| const tnsr::iiaa<double, 3>& dij_metric, | ||
| const tnsr::AA<double, 3>& inverse_metric, | ||
| const tnsr::iAA<double, 3>& di_inverse_metric) { | ||
| tnsr::iAbb<double, 3> di_christoffel{}; | ||
| tnsr::abb<double, 3> d_metric{}; | ||
| tnsr::iabb<double, 3> di_d_metric{}; | ||
| for (size_t a = 0; a <= 3; ++a) { | ||
| for (size_t b = 0; b <= 3; ++b) { | ||
| d_metric.get(0, a, b) = 0.; | ||
| for (size_t i = 0; i < 3; ++i) { | ||
| d_metric.get(i + 1, a, b) = di_metric.get(i, a, b); | ||
| di_d_metric.get(i, 0, a, b) = 0.; | ||
| for (size_t j = 0; j < 3; ++j) { | ||
| di_d_metric.get(i, j + 1, a, b) = dij_metric.get(i, j, a, b); | ||
| } | ||
| } | ||
| } | ||
| } | ||
| tenex::evaluate<ti::i, ti::A, ti::b, ti::c>( | ||
| make_not_null(&di_christoffel), | ||
| 0.5 * di_inverse_metric(ti::i, ti::A, ti::D) * | ||
| (d_metric(ti::b, ti::c, ti::d) + d_metric(ti::c, ti::b, ti::d) - | ||
| d_metric(ti::d, ti::b, ti::c)) + | ||
| 0.5 * inverse_metric(ti::A, ti::D) * | ||
| (di_d_metric(ti::i, ti::b, ti::c, ti::d) + | ||
| di_d_metric(ti::i, ti::c, ti::b, ti::d) - | ||
| di_d_metric(ti::i, ti::d, ti::b, ti::c))); | ||
| return di_christoffel; | ||
| } | ||
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| tnsr::iA<double, 3> spatial_derivative_ks_contracted_christoffel( | ||
| const tnsr::I<double, 3>& pos) { | ||
| const double r_sq = get(dot_product(pos, pos)); | ||
| const double r = sqrt(r_sq); | ||
| const double one_over_r = 1. / r; | ||
| const double one_over_r_2 = 1. / r_sq; | ||
| const double one_over_r_3 = cube(one_over_r); | ||
| const double one_over_r_4 = square(one_over_r_2); | ||
| const double one_over_r_5 = one_over_r_4 * one_over_r; | ||
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| tnsr::iA<double, 3> di_contracted_christoffel{}; | ||
| for (size_t i = 0; i < 3; ++i) { | ||
| di_contracted_christoffel.get(i, 0) = 4. * pos.get(i) * one_over_r_4; | ||
| for (size_t j = 0; j < 3; ++j) { | ||
| di_contracted_christoffel.get(i, j + 1) = | ||
| -6. * pos.get(i) * pos.get(j) * one_over_r_5; | ||
| } | ||
| di_contracted_christoffel.get(i, i + 1) += 2. * one_over_r_3; | ||
| } | ||
| return di_contracted_christoffel; | ||
| } | ||
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| } // namespace CurvedScalarWave::Worldtube |
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src/Evolution/Systems/CurvedScalarWave/Worldtube/KerrSchildDerivatives.hpp
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,64 @@ | ||
| // Distributed under the MIT License. | ||
| // See LICENSE.txt for details. | ||
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| #pragma once | ||
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| #include <cstddef> | ||
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| #include "DataStructures/Tensor/Tensor.hpp" | ||
| #include "Utilities/Gsl.hpp" | ||
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| namespace CurvedScalarWave::Worldtube { | ||
| /*! | ||
| * \brief The spatial derivative of the zero spin inverse Kerr Schild metric, | ||
| * $\partial_i g^{\mu \nu}$, assuming a black hole at the coordinate center with | ||
| * mass M = 1. | ||
| */ | ||
| tnsr::iAA<double, 3> spatial_derivative_inverse_ks_metric( | ||
| const tnsr::I<double, 3>& pos); | ||
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| /*! | ||
| * \brief The spatial derivative of the spacetime metric, | ||
| * $\partial_i g_{\mu \nu}$. | ||
| */ | ||
| tnsr::iaa<double, 3> spatial_derivative_ks_metric( | ||
| const tnsr::aa<double, 3>& metric, | ||
| const tnsr::iAA<double, 3>& di_inverse_metric); | ||
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| /*! | ||
| * \brief The second spatial derivative of the zero spin inverse Kerr Schild | ||
| * metric, $\partial_i \partial_j g^{\mu \nu}$, assuming a black hole at the | ||
| * coordinate center with mass M = 1. | ||
| */ | ||
| tnsr::iiAA<double, 3> second_spatial_derivative_inverse_ks_metric( | ||
| const tnsr::I<double, 3>& pos); | ||
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| /*! | ||
| * \brief The spatial derivative of the spacetime metric, | ||
| * $\partial_i \partial_j g_{\mu \nu}$. | ||
| */ | ||
| tnsr::iiaa<double, 3> second_spatial_derivative_metric( | ||
| const tnsr::aa<double, 3>& metric, const tnsr::iaa<double, 3>& di_metric, | ||
| const tnsr::iAA<double, 3>& di_inverse_metric, | ||
| const tnsr::iiAA<double, 3>& dij_inverse_metric); | ||
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| /*! | ||
| * \brief The spatial derivative of the Christoffel | ||
| * symbols, $\partial_i \Gamma^\rho_{\mu \nu}$. | ||
| */ | ||
| tnsr::iAbb<double, 3> spatial_derivative_christoffel( | ||
| const tnsr::iaa<double, 3>& di_metric, | ||
| const tnsr::iiaa<double, 3>& dij_metric, | ||
| const tnsr::AA<double, 3>& inverse_metric, | ||
| const tnsr::iAA<double, 3>& di_inverse_metric); | ||
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| /*! | ||
| * \brief The spatial derivative of the zero spin Kerr Schild contracted | ||
| * Christoffel symbols, | ||
| * $\partial_i g^{\mu \nu} \Gamma^\rho_{\mu \nu}$, assuming a black hole at the | ||
| * coordinate center with mass M = 1. | ||
| */ | ||
| tnsr::iA<double, 3> spatial_derivative_ks_contracted_christoffel( | ||
| const tnsr::I<double, 3>& pos); | ||
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| } // namespace CurvedScalarWave::Worldtube |
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