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SpECTRE
v2024.04.12
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Holds functions related to general relativity. More...
Namespaces | |
| namespace | AnalyticData |
| Classes which implement analytic data for general relativity. | |
| namespace | Solutions |
| Classes which implement analytic solutions to Einstein's equations. | |
Classes | |
| struct | AnalyticDataBase |
| Base struct for properties common to all GR analytic data classes. More... | |
| struct | AnalyticSolution |
| Base struct for properties common to all GR analytic solutions. More... | |
| class | KerrSchildCoords |
| Contains helper functions for transforming tensors in Kerr spacetime to Kerr-Schild coordinates. More... | |
Typedefs | |
| template<size_t Dim, typename DataType > | |
| using | tags_for_hydro = tmpl::list< gr::Tags::Lapse< DataType >, gr::Tags::Shift< DataType, Dim >, gr::Tags::SpatialMetric< DataType, Dim >, gr::Tags::InverseSpatialMetric< DataType, Dim >, gr::Tags::SqrtDetSpatialMetric< DataType >, ::Tags::deriv< gr::Tags::Lapse< DataType >, tmpl::size_t< Dim >, Frame::Inertial >, ::Tags::deriv< gr::Tags::Shift< DataType, Dim >, tmpl::size_t< Dim >, Frame::Inertial >, ::Tags::deriv< gr::Tags::SpatialMetric< DataType, Dim >, tmpl::size_t< Dim >, Frame::Inertial >, gr::Tags::ExtrinsicCurvature< DataType, Dim > > |
| GR Tags commonly needed for the evolution of hydro systems. | |
| template<size_t Dim, typename DataType > | |
| using | analytic_solution_tags = tmpl::list< gr::Tags::Lapse< DataType >, ::Tags::dt< gr::Tags::Lapse< DataType > >, ::Tags::deriv< gr::Tags::Lapse< DataType >, tmpl::size_t< Dim >, Frame::Inertial >, gr::Tags::Shift< DataType, Dim >, ::Tags::dt< gr::Tags::Shift< DataType, Dim > >, ::Tags::deriv< gr::Tags::Shift< DataType, Dim >, tmpl::size_t< Dim >, Frame::Inertial >, gr::Tags::SpatialMetric< DataType, Dim >, ::Tags::dt< gr::Tags::SpatialMetric< DataType, Dim > >, ::Tags::deriv< gr::Tags::SpatialMetric< DataType, Dim >, tmpl::size_t< Dim >, Frame::Inertial >, gr::Tags::SqrtDetSpatialMetric< DataType >, gr::Tags::ExtrinsicCurvature< DataType, Dim >, gr::Tags::InverseSpatialMetric< DataType, Dim > > |
| The tags for the variables returned by GR analytic solutions. | |
Functions | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| tnsr::A< DataType, VolumeDim, Frame > | interface_null_normal (const tnsr::A< DataType, VolumeDim, Frame > &spacetime_normal_vector, const tnsr::I< DataType, VolumeDim, Frame > &interface_unit_normal_vector, const double sign) |
| Compute null normal vector to the boundary of a closed region in a spatial slice of spacetime. More... | |
| bool | operator!= (const KerrSchildCoords &lhs, const KerrSchildCoords &rhs) |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| tnsr::ii< DataType, VolumeDim, Frame > | transverse_projection_operator (const tnsr::ii< DataType, VolumeDim, Frame > &spatial_metric, const tnsr::i< DataType, VolumeDim, Frame > &normal_one_form) |
| Compute projection operator onto an interface. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| tnsr::Ij< DataType, VolumeDim, Frame > | transverse_projection_operator (const tnsr::I< DataType, VolumeDim, Frame > &normal_vector, const tnsr::i< DataType, VolumeDim, Frame > &normal_one_form) |
| Compute projection operator onto an interface. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| tnsr::AA< DataType, VolumeDim, Frame > | transverse_projection_operator (const tnsr::AA< DataType, VolumeDim, Frame > &inverse_spacetime_metric, const tnsr::A< DataType, VolumeDim, Frame > &spacetime_normal_vector, const tnsr::I< DataType, VolumeDim, Frame > &interface_unit_normal_vector) |
| Compute spacetime projection operator onto an interface. More... | |
| template<size_t SpatialDim, typename Frame , IndexType Index, typename DataType > | |
| void | christoffel_first_kind (gsl::not_null< tnsr::abb< DataType, SpatialDim, Frame, Index > * > christoffel, const tnsr::abb< DataType, SpatialDim, Frame, Index > &d_metric) |
| Computes Christoffel symbol of the first kind from derivative of metric. More... | |
| template<size_t SpatialDim, typename Frame , IndexType Index, typename DataType > | |
| tnsr::abb< DataType, SpatialDim, Frame, Index > | christoffel_first_kind (const tnsr::abb< DataType, SpatialDim, Frame, Index > &d_metric) |
| Computes Christoffel symbol of the first kind from derivative of metric. More... | |
| template<size_t SpatialDim, typename Frame , IndexType Index, typename DataType > | |
| void | christoffel_second_kind (const gsl::not_null< tnsr::Abb< DataType, SpatialDim, Frame, Index > * > christoffel, const tnsr::abb< DataType, SpatialDim, Frame, Index > &d_metric, const tnsr::AA< DataType, SpatialDim, Frame, Index > &inverse_metric) |
| Computes Christoffel symbol of the second kind from derivative of metric and the inverse metric. More... | |
| template<size_t SpatialDim, typename Frame , IndexType Index, typename DataType > | |
| auto | christoffel_second_kind (const tnsr::abb< DataType, SpatialDim, Frame, Index > &d_metric, const tnsr::AA< DataType, SpatialDim, Frame, Index > &inverse_metric) -> tnsr::Abb< DataType, SpatialDim, Frame, Index > |
| Computes Christoffel symbol of the second kind from derivative of metric and the inverse metric. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | derivatives_of_spacetime_metric (gsl::not_null< tnsr::abb< DataType, SpatialDim, Frame > * > spacetime_deriv_spacetime_metric, const Scalar< DataType > &lapse, const Scalar< DataType > &dt_lapse, const tnsr::i< DataType, SpatialDim, Frame > &deriv_lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::I< DataType, SpatialDim, Frame > &dt_shift, const tnsr::iJ< DataType, SpatialDim, Frame > &deriv_shift, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::ii< DataType, SpatialDim, Frame > &dt_spatial_metric, const tnsr::ijj< DataType, SpatialDim, Frame > &deriv_spatial_metric) |
| Computes spacetime derivative of spacetime metric from spatial metric, lapse, shift, and their space and time derivatives. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::abb< DataType, SpatialDim, Frame > | derivatives_of_spacetime_metric (const Scalar< DataType > &lapse, const Scalar< DataType > &dt_lapse, const tnsr::i< DataType, SpatialDim, Frame > &deriv_lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::I< DataType, SpatialDim, Frame > &dt_shift, const tnsr::iJ< DataType, SpatialDim, Frame > &deriv_shift, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::ii< DataType, SpatialDim, Frame > &dt_spatial_metric, const tnsr::ijj< DataType, SpatialDim, Frame > &deriv_spatial_metric) |
| Computes spacetime derivative of spacetime metric from spatial metric, lapse, shift, and their space and time derivatives. More... | |
| template<typename DataType , size_t Dim, typename Frame > | |
| void | deriv_inverse_spatial_metric (const gsl::not_null< tnsr::iJJ< DataType, Dim, Frame > * > result, const tnsr::II< DataType, Dim, Frame > &inverse_spatial_metric, const tnsr::ijj< DataType, Dim, Frame > &d_spatial_metric) |
| Computes the spatial derivative of the inverse spatial metric from the inverse spatial metric and the spatial derivative of the spatial metric. More... | |
| template<typename DataType , size_t Dim, typename Frame > | |
| tnsr::iJJ< DataType, Dim, Frame > | deriv_inverse_spatial_metric (const tnsr::II< DataType, Dim, Frame > &inverse_spatial_metric, const tnsr::ijj< DataType, Dim, Frame > &d_spatial_metric) |
| Computes the spatial derivative of the inverse spatial metric from the inverse spatial metric and the spatial derivative of the spatial metric. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::ii< DataType, SpatialDim, Frame > | extrinsic_curvature (const Scalar< DataType > &lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::iJ< DataType, SpatialDim, Frame > &deriv_shift, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::ii< DataType, SpatialDim, Frame > &dt_spatial_metric, const tnsr::ijj< DataType, SpatialDim, Frame > &deriv_spatial_metric) |
| Computes extrinsic curvature from metric and derivatives. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | extrinsic_curvature (gsl::not_null< tnsr::ii< DataType, SpatialDim, Frame > * > ex_curvature, const Scalar< DataType > &lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::iJ< DataType, SpatialDim, Frame > &deriv_shift, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::ii< DataType, SpatialDim, Frame > &dt_spatial_metric, const tnsr::ijj< DataType, SpatialDim, Frame > &deriv_spatial_metric) |
| Computes extrinsic curvature from metric and derivatives. More... | |
| template<typename DataType , size_t Dim> | |
| void | geodesic_acceleration (gsl::not_null< tnsr::I< DataType, Dim > * > acceleration, const tnsr::I< DataType, Dim > &velocity, const tnsr::Abb< DataType, Dim > &christoffel_second_kind) |
| Computes the coordinate geodesic acceleration in the inertial frame. More... | |
| template<typename DataType , size_t Dim> | |
| tnsr::I< DataType, Dim > | geodesic_acceleration (const tnsr::I< DataType, Dim > &velocity, const tnsr::Abb< DataType, Dim > &christoffel_second_kind) |
| Computes the coordinate geodesic acceleration in the inertial frame. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| tnsr::a< DataType, VolumeDim, Frame > | interface_null_normal (const tnsr::a< DataType, VolumeDim, Frame > &spacetime_normal_one_form, const tnsr::i< DataType, VolumeDim, Frame > &interface_unit_normal_one_form, const double sign) |
| Compute null normal one-form to the boundary of a closed region in a spatial slice of spacetime. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| void | interface_null_normal (gsl::not_null< tnsr::A< DataType, VolumeDim, Frame > * > null_vector, const tnsr::A< DataType, VolumeDim, Frame > &spacetime_normal_vector, const tnsr::I< DataType, VolumeDim, Frame > &interface_unit_normal_vector, const double sign) |
| Compute null normal vector to the boundary of a closed region in a spatial slice of spacetime. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | inverse_spacetime_metric (gsl::not_null< tnsr::AA< DataType, SpatialDim, Frame > * > inverse_spacetime_metric, const Scalar< DataType > &lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Compute inverse spacetime metric from inverse spatial metric, lapse and shift. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::AA< DataType, SpatialDim, Frame > | inverse_spacetime_metric (const Scalar< DataType > &lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Compute inverse spacetime metric from inverse spatial metric, lapse and shift. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| Scalar< DataType > | lapse (const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::aa< DataType, SpatialDim, Frame > &spacetime_metric) |
| Compute lapse from shift and spacetime metric. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | lapse (gsl::not_null< Scalar< DataType > * > lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::aa< DataType, SpatialDim, Frame > &spacetime_metric) |
| Compute lapse from shift and spacetime metric. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| tnsr::II< DataType, VolumeDim, Frame > | transverse_projection_operator (const tnsr::II< DataType, VolumeDim, Frame > &inverse_spatial_metric, const tnsr::I< DataType, VolumeDim, Frame > &normal_vector) |
| Compute projection operator onto an interface. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| void | transverse_projection_operator (gsl::not_null< tnsr::ii< DataType, VolumeDim, Frame > * > projection_tensor, const tnsr::ii< DataType, VolumeDim, Frame > &spatial_metric, const tnsr::i< DataType, VolumeDim, Frame > &normal_one_form) |
| Compute projection operator onto an interface. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| tnsr::aa< DataType, VolumeDim, Frame > | transverse_projection_operator (const tnsr::aa< DataType, VolumeDim, Frame > &spacetime_metric, const tnsr::a< DataType, VolumeDim, Frame > &spacetime_normal_one_form, const tnsr::i< DataType, VolumeDim, Frame > &interface_unit_normal_one_form) |
| Compute spacetime projection operator onto an interface. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| void | transverse_projection_operator (gsl::not_null< tnsr::AA< DataType, VolumeDim, Frame > * > projection_tensor, const tnsr::AA< DataType, VolumeDim, Frame > &inverse_spacetime_metric, const tnsr::A< DataType, VolumeDim, Frame > &spacetime_normal_vector, const tnsr::I< DataType, VolumeDim, Frame > &interface_unit_normal_vector) |
| Compute spacetime projection operator onto an interface. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| tnsr::Ab< DataType, VolumeDim, Frame > | transverse_projection_operator (const tnsr::A< DataType, VolumeDim, Frame > &spacetime_normal_vector, const tnsr::a< DataType, VolumeDim, Frame > &spacetime_normal_one_form, const tnsr::I< DataType, VolumeDim, Frame > &interface_unit_normal_vector, const tnsr::i< DataType, VolumeDim, Frame > &interface_unit_normal_one_form) |
| Compute spacetime projection operator onto an interface. More... | |
| template<typename DataType , size_t VolumeDim, typename Frame > | |
| void | transverse_projection_operator (gsl::not_null< tnsr::Ab< DataType, VolumeDim, Frame > * > projection_tensor, const tnsr::A< DataType, VolumeDim, Frame > &spacetime_normal_vector, const tnsr::a< DataType, VolumeDim, Frame > &spacetime_normal_one_form, const tnsr::I< DataType, VolumeDim, Frame > &interface_unit_normal_vector, const tnsr::i< DataType, VolumeDim, Frame > &interface_unit_normal_one_form) |
| Compute spacetime projection operator onto an interface. More... | |
| template<typename Frame > | |
| void | psi_4 (const gsl::not_null< Scalar< ComplexDataVector > * > psi_4_result, const tnsr::ii< DataVector, 3, Frame > &spatial_ricci, const tnsr::ii< DataVector, 3, Frame > &extrinsic_curvature, const tnsr::ijj< DataVector, 3, Frame > &cov_deriv_extrinsic_curvature, const tnsr::ii< DataVector, 3, Frame > &spatial_metric, const tnsr::II< DataVector, 3, Frame > &inverse_spatial_metric, const tnsr::I< DataVector, 3, Frame > &inertial_coords) |
| Computes Newman Penrose quantity \(\Psi_4\) using the characteristic field U \(^{8+}\) and complex vector \(\bar{m}^i\). More... | |
| template<typename Frame > | |
| Scalar< ComplexDataVector > | psi_4 (const tnsr::ii< DataVector, 3, Frame > &spatial_ricci, const tnsr::ii< DataVector, 3, Frame > &extrinsic_curvature, const tnsr::ijj< DataVector, 3, Frame > &cov_deriv_extrinsic_curvature, const tnsr::ii< DataVector, 3, Frame > &spatial_metric, const tnsr::II< DataVector, 3, Frame > &inverse_spatial_metric, const tnsr::I< DataVector, 3, Frame > &inertial_coords) |
| Computes Newman Penrose quantity \(\Psi_4\) using the characteristic field U \(^{8+}\) and complex vector \(\bar{m}^i\). More... | |
| template<typename Frame > | |
| void | psi_4_real (const gsl::not_null< Scalar< DataVector > * > psi_4_real_result, const tnsr::ii< DataVector, 3, Frame > &spatial_ricci, const tnsr::ii< DataVector, 3, Frame > &extrinsic_curvature, const tnsr::ijj< DataVector, 3, Frame > &cov_deriv_extrinsic_curvature, const tnsr::ii< DataVector, 3, Frame > &spatial_metric, const tnsr::II< DataVector, 3, Frame > &inverse_spatial_metric, const tnsr::I< DataVector, 3, Frame > &inertial_coords) |
| Computes the real part of the Newman Penrose quantity \(\Psi_4\) using \(\Psi_4[Real] = -0.5*U^{8+}_{ij}*(x^ix^j - y^iy^j)\). | |
| template<typename Frame > | |
| Scalar< DataVector > | psi_4_real (const tnsr::ii< DataVector, 3, Frame > &spatial_ricci, const tnsr::ii< DataVector, 3, Frame > &extrinsic_curvature, const tnsr::ijj< DataVector, 3, Frame > &cov_deriv_extrinsic_curvature, const tnsr::ii< DataVector, 3, Frame > &spatial_metric, const tnsr::II< DataVector, 3, Frame > &inverse_spatial_metric, const tnsr::I< DataVector, 3, Frame > &inertial_coords) |
| Computes the real part of the Newman Penrose quantity \(\Psi_4\) using \(\Psi_4[Real] = -0.5*U^{8+}_{ij}*(x^ix^j - y^iy^j)\). | |
| template<size_t SpatialDim, typename Frame , IndexType Index, typename DataType > | |
| void | ricci_tensor (gsl::not_null< tnsr::aa< DataType, SpatialDim, Frame, Index > * > result, const tnsr::Abb< DataType, SpatialDim, Frame, Index > &christoffel_2nd_kind, const tnsr::aBcc< DataType, SpatialDim, Frame, Index > &d_christoffel_2nd_kind) |
| Computes Ricci tensor from the (spatial or spacetime) Christoffel symbol of the second kind and its derivative. More... | |
| 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) |
| Computes Ricci tensor from the (spatial or spacetime) Christoffel symbol of the second kind and its derivative. More... | |
| template<size_t SpatialDim, typename Frame , IndexType Index, typename DataType > | |
| void | ricci_scalar (const gsl::not_null< Scalar< DataType > * > ricci_scalar_result, const tnsr::aa< DataType, SpatialDim, Frame, Index > &ricci_tensor, const tnsr::AA< DataType, SpatialDim, Frame, Index > &inverse_metric) |
| Computes the Ricci Scalar from the (spatial or spacetime) Ricci Tensor and inverse metrics. More... | |
| template<size_t SpatialDim, typename Frame , IndexType Index, typename DataType > | |
| Scalar< DataType > | ricci_scalar (const tnsr::aa< DataType, SpatialDim, Frame, Index > &ricci_tensor, const tnsr::AA< DataType, SpatialDim, Frame, Index > &inverse_metric) |
| Computes the Ricci Scalar from the (spatial or spacetime) Ricci Tensor and inverse metrics. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::I< DataType, SpatialDim, Frame > | shift (const tnsr::aa< DataType, SpatialDim, Frame > &spacetime_metric, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Compute shift from spacetime metric and inverse spatial metric. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | shift (gsl::not_null< tnsr::I< DataType, SpatialDim, Frame > * > shift, const tnsr::aa< DataType, SpatialDim, Frame > &spacetime_metric, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Compute shift from spacetime metric and inverse spatial metric. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | spacetime_deriv_of_goth_g (gsl::not_null< tnsr::aBB< DataType, SpatialDim, Frame > * > da_goth_g, const tnsr::AA< DataType, SpatialDim, Frame > &inverse_spacetime_metric, const tnsr::abb< DataType, SpatialDim, Frame > &da_spacetime_metric, const Scalar< DataType > &lapse, const tnsr::a< DataType, SpatialDim, Frame > &da_lapse, const Scalar< DataType > &sqrt_det_spatial_metric, const tnsr::a< DataType, SpatialDim, Frame > &da_det_spatial_metric) |
| Computes spacetime derivative of \( \mathfrak{g}^{ab}\equiv (-g)^{1/2} g^{ab} \). More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::aBB< DataType, SpatialDim, Frame > | spacetime_deriv_of_goth_g (const tnsr::AA< DataType, SpatialDim, Frame > &inverse_spacetime_metric, const tnsr::abb< DataType, SpatialDim, Frame > &da_spacetime_metric, const Scalar< DataType > &lapse, const tnsr::a< DataType, SpatialDim, Frame > &da_lapse, const Scalar< DataType > &sqrt_det_spatial_metric, const tnsr::a< DataType, SpatialDim, Frame > &da_det_spatial_metric) |
| Computes spacetime derivative of \( \mathfrak{g}^{ab}\equiv (-g)^{1/2} g^{ab} \). More... | |
| template<typename DataType , size_t Dim, typename Frame > | |
| void | spacetime_metric (gsl::not_null< tnsr::aa< DataType, Dim, Frame > * > spacetime_metric, const Scalar< DataType > &lapse, const tnsr::I< DataType, Dim, Frame > &shift, const tnsr::ii< DataType, Dim, Frame > &spatial_metric) |
| Computes the spacetime metric from the spatial metric, lapse, and shift. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::aa< DataType, SpatialDim, Frame > | spacetime_metric (const Scalar< DataType > &lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric) |
| Computes the spacetime metric from the spatial metric, lapse, and shift. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | spacetime_normal_one_form (gsl::not_null< tnsr::a< DataType, SpatialDim, Frame > * > normal_one_form, const Scalar< DataType > &lapse) |
| Computes spacetime normal one-form from lapse. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::a< DataType, SpatialDim, Frame > | spacetime_normal_one_form (const Scalar< DataType > &lapse) |
| Computes spacetime normal one-form from lapse. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::A< DataType, SpatialDim, Frame > | spacetime_normal_vector (const Scalar< DataType > &lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift) |
| Computes spacetime normal vector from lapse and shift. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | spacetime_normal_vector (gsl::not_null< tnsr::A< DataType, SpatialDim, Frame > * > spacetime_normal_vector, const Scalar< DataType > &lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift) |
| Computes spacetime normal vector from lapse and shift. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::ii< DataType, SpatialDim, Frame > | spatial_metric (const tnsr::aa< DataType, SpatialDim, Frame > &spacetime_metric) |
| Compute spatial metric from spacetime metric. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | spatial_metric (gsl::not_null< tnsr::ii< DataType, SpatialDim, Frame > * > spatial_metric, const tnsr::aa< DataType, SpatialDim, Frame > &spacetime_metric) |
| Compute spatial metric from spacetime metric. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | time_derivative_of_spacetime_metric (gsl::not_null< tnsr::aa< DataType, SpatialDim, Frame > * > dt_spacetime_metric, const Scalar< DataType > &lapse, const Scalar< DataType > &dt_lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::I< DataType, SpatialDim, Frame > &dt_shift, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::ii< DataType, SpatialDim, Frame > &dt_spatial_metric) |
| Computes the time derivative of the spacetime metric from spatial metric, lapse, shift, and their time derivatives. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::aa< DataType, SpatialDim, Frame > | time_derivative_of_spacetime_metric (const Scalar< DataType > &lapse, const Scalar< DataType > &dt_lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::I< DataType, SpatialDim, Frame > &dt_shift, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::ii< DataType, SpatialDim, Frame > &dt_spatial_metric) |
| Computes the time derivative of the spacetime metric from spatial metric, lapse, shift, and their time derivatives. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | time_derivative_of_spatial_metric (gsl::not_null< tnsr::ii< DataType, SpatialDim, Frame > * > dt_spatial_metric, const Scalar< DataType > &lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::iJ< DataType, SpatialDim, Frame > &deriv_shift, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::ijj< DataType, SpatialDim, Frame > &deriv_spatial_metric, const tnsr::ii< DataType, SpatialDim, Frame > &extrinsic_curvature) |
| Computes the time derivative of the spatial metric from extrinsic curvature, lapse, shift, and their time derivatives. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::ii< DataType, SpatialDim, Frame > | time_derivative_of_spatial_metric (const Scalar< DataType > &lapse, const tnsr::I< DataType, SpatialDim, Frame > &shift, const tnsr::iJ< DataType, SpatialDim, Frame > &deriv_shift, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::ijj< DataType, SpatialDim, Frame > &deriv_spatial_metric, const tnsr::ii< DataType, SpatialDim, Frame > &extrinsic_curvature) |
| Computes the time derivative of the spatial metric from extrinsic curvature, lapse, shift, and their time derivatives. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::ii< DataType, SpatialDim, Frame > | weyl_electric (const tnsr::ii< DataType, SpatialDim, Frame > &spatial_ricci, const tnsr::ii< DataType, SpatialDim, Frame > &extrinsic_curvature, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Computes the electric part of the Weyl tensor in vacuum. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | weyl_electric (gsl::not_null< tnsr::ii< DataType, SpatialDim, Frame > * > weyl_electric_part, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_ricci, const tnsr::ii< DataType, SpatialDim, Frame > &extrinsic_curvature, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Computes the electric part of the Weyl tensor in vacuum. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| Scalar< DataType > | weyl_electric_scalar (const tnsr::ii< DataType, SpatialDim, Frame > &weyl_electric, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Computes the scalar \(E_{ij} E^{ij}\) from the electric part of the Weyl tensor \(E_{ij}\). More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | weyl_electric_scalar (gsl::not_null< Scalar< DataType > * > weyl_electric_scalar_result, const tnsr::ii< DataType, SpatialDim, Frame > &weyl_electric, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Computes the scalar \(E_{ij} E^{ij}\) from the electric part of the Weyl tensor \(E_{ij}\). More... | |
| template<typename Frame , typename DataType > | |
| tnsr::ii< DataType, 3, Frame > | weyl_magnetic (const tnsr::ijj< DataType, 3, Frame > &grad_extrinsic_curvature, const tnsr::ii< DataType, 3, Frame > &spatial_metric, const Scalar< DataType > &sqrt_det_spatial_metric) |
| Computes the magnetic part of the Weyl tensor. More... | |
| template<typename Frame , typename DataType > | |
| void | weyl_magnetic (gsl::not_null< tnsr::ii< DataType, 3, Frame > * > weyl_magnetic_part, const tnsr::ijj< DataType, 3, Frame > &grad_extrinsic_curvature, const tnsr::ii< DataType, 3, Frame > &spatial_metric, const Scalar< DataType > &sqrt_det_spatial_metric) |
| Computes the magnetic part of the Weyl tensor. More... | |
| template<typename Frame , typename DataType > | |
| Scalar< DataType > | weyl_magnetic_scalar (const tnsr::ii< DataType, 3, Frame > &weyl_magnetic, const tnsr::II< DataType, 3, Frame > &inverse_spatial_metric) |
| Computes the scalar \(B_{ij} B^{ij}\) from the magnetic part of the Weyl tensor \(B_{ij}\). More... | |
| template<typename Frame , typename DataType > | |
| void | weyl_magnetic_scalar (gsl::not_null< Scalar< DataType > * > weyl_magnetic_scalar_result, const tnsr::ii< DataType, 3, Frame > &weyl_magnetic, const tnsr::II< DataType, 3, Frame > &inverse_spatial_metric) |
| Computes the scalar \(B_{ij} B^{ij}\) from the magnetic part of the Weyl tensor \(B_{ij}\). More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::ii< DataType, SpatialDim, Frame > | weyl_propagating (const tnsr::ii< DataType, SpatialDim, Frame > &ricci, const tnsr::ii< DataType, SpatialDim, Frame > &extrinsic_curvature, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric, const tnsr::ijj< DataType, SpatialDim, Frame > &cov_deriv_extrinsic_curvature, const tnsr::I< DataType, SpatialDim, Frame > &unit_interface_normal_vector, const tnsr::II< DataType, SpatialDim, Frame > &projection_IJ, const tnsr::ii< DataType, SpatialDim, Frame > &projection_ij, const tnsr::Ij< DataType, SpatialDim, Frame > &projection_Ij, const double sign) |
| Computes the propagating modes of the Weyl tensor. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | weyl_propagating (gsl::not_null< tnsr::ii< DataType, SpatialDim, Frame > * > weyl_prop_u8, const tnsr::ii< DataType, SpatialDim, Frame > &ricci, const tnsr::ii< DataType, SpatialDim, Frame > &extrinsic_curvature, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric, const tnsr::ijj< DataType, SpatialDim, Frame > &cov_deriv_extrinsic_curvature, const tnsr::I< DataType, SpatialDim, Frame > &unit_interface_normal_vector, const tnsr::II< DataType, SpatialDim, Frame > &projection_IJ, const tnsr::ii< DataType, SpatialDim, Frame > &projection_ij, const tnsr::Ij< DataType, SpatialDim, Frame > &projection_Ij, const double sign) |
| Computes the propagating modes of the Weyl tensor. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| tnsr::ii< DataType, SpatialDim, Frame > | weyl_type_D1 (const tnsr::ii< DataType, SpatialDim, Frame > &weyl_electric, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Computes a quantity measuring how far from type D spacetime is. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | weyl_type_D1 (const gsl::not_null< tnsr::ii< DataType, SpatialDim, Frame > * > weyl_type_D1, const tnsr::ii< DataType, SpatialDim, Frame > &weyl_electric, const tnsr::ii< DataType, SpatialDim, Frame > &spatial_metric, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Computes a quantity measuring how far from type D spacetime is. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| void | weyl_type_D1_scalar (const gsl::not_null< Scalar< DataType > * > weyl_type_D1_scalar_result, const tnsr::ii< DataType, SpatialDim, Frame > &weyl_type_D1, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Computes the scalar \(D_{ij} D^{ij}\) , a measure of a spacetime's devitation from type D. More... | |
| template<typename DataType , size_t SpatialDim, typename Frame > | |
| Scalar< DataType > | weyl_type_D1_scalar (const tnsr::ii< DataType, SpatialDim, Frame > &weyl_type_D1, const tnsr::II< DataType, SpatialDim, Frame > &inverse_spatial_metric) |
| Computes the scalar \(D_{ij} D^{ij}\) , a measure of a spacetime's devitation from type D. More... | |
Holds functions related to general relativity.
| tnsr::I< DataType, Dim > gr::geodesic_acceleration | ( | const tnsr::I< DataType, Dim > & | velocity, |
| const tnsr::Abb< DataType, Dim > & | christoffel_second_kind | ||
| ) |
Computes the coordinate geodesic acceleration in the inertial frame.
The geodesic acceleration in coordinate form is given by
\begin{equation} \frac{d^2 x^i}{d t^2} = (v^i \Gamma^0_{00} - \Gamma^i_{00} ) + 2 v^j (v^i \Gamma^0_{j0} - \Gamma^i_{j0} ) + v^j v^k (v^i \Gamma^0_{jk} - \Gamma^i_{jk} ), \end{equation}
where \(v^i\) is the coordinate velocity, \(\Gamma^\mu_{\nu \rho}\) are the spacetime Christoffel symbols of the second kind, and all latin indices are spatial.
| void gr::geodesic_acceleration | ( | gsl::not_null< tnsr::I< DataType, Dim > * > | acceleration, |
| const tnsr::I< DataType, Dim > & | velocity, | ||
| const tnsr::Abb< DataType, Dim > & | christoffel_second_kind | ||
| ) |
Computes the coordinate geodesic acceleration in the inertial frame.
The geodesic acceleration in coordinate form is given by
\begin{equation} \frac{d^2 x^i}{d t^2} = (v^i \Gamma^0_{00} - \Gamma^i_{00} ) + 2 v^j (v^i \Gamma^0_{j0} - \Gamma^i_{j0} ) + v^j v^k (v^i \Gamma^0_{jk} - \Gamma^i_{jk} ), \end{equation}
where \(v^i\) is the coordinate velocity, \(\Gamma^\mu_{\nu \rho}\) are the spacetime Christoffel symbols of the second kind, and all latin indices are spatial.
| tnsr::A< DataType, VolumeDim, Frame > gr::interface_null_normal | ( | const tnsr::A< DataType, VolumeDim, Frame > & | spacetime_normal_vector, |
| const tnsr::I< DataType, VolumeDim, Frame > & | interface_unit_normal_vector, | ||
| const double | sign | ||
| ) |
Compute null normal vector to the boundary of a closed region in a spatial slice of spacetime.
Consider an \(n-1\)-dimensional boundary \(S\) of a closed region in an \(n\)-dimensional spatial hypersurface \(\Sigma\). Let \(s^a\) be the unit spacelike vector orthogonal to \(S\) in \(\Sigma\), and \(n^a\) be the timelike unit vector orthogonal to \(\Sigma\). This function returns the null vector that is outgoing/ingoing on \(S\):
\begin{align*} k^a = \frac{1}{\sqrt{2}}\left(n^a \pm s^a\right). \end{align*}
| tnsr::a< DataType, SpatialDim, Frame > gr::spacetime_normal_one_form | ( | const Scalar< DataType > & | lapse | ) |
Computes spacetime normal one-form from lapse.
If \(\alpha\) is the lapse, then
\begin{align} n_t &= - \alpha \\ n_i &= 0 \end{align}
is computed.
| void gr::spacetime_normal_one_form | ( | gsl::not_null< tnsr::a< DataType, SpatialDim, Frame > * > | normal_one_form, |
| const Scalar< DataType > & | lapse | ||
| ) |
Computes spacetime normal one-form from lapse.
If \(\alpha\) is the lapse, then
\begin{align} n_t &= - \alpha \\ n_i &= 0 \end{align}
is computed.
| tnsr::AA< DataType, VolumeDim, Frame > gr::transverse_projection_operator | ( | const tnsr::AA< DataType, VolumeDim, Frame > & | inverse_spacetime_metric, |
| const tnsr::A< DataType, VolumeDim, Frame > & | spacetime_normal_vector, | ||
| const tnsr::I< DataType, VolumeDim, Frame > & | interface_unit_normal_vector | ||
| ) |
Compute spacetime projection operator onto an interface.
Consider a \(d-1\)-dimensional surface \(S\) in a \(d\)-dimensional spatial hypersurface \(\Sigma\). Let \(s^a\) be the unit spacelike vector orthogonal to \(S\) in \(\Sigma\), and \(n^a\) be the timelike unit vector orthogonal to \(\Sigma\). This function returns the projection operator onto \(S\) for \(d+1\) dimensional quantities:
\begin{align*} P^{ab} = g^{ab} + n^a n^b - s^a s^b = \gamma_{ab} - s_a s_b. \end{align*}
| tnsr::Ij< DataType, VolumeDim, Frame > gr::transverse_projection_operator | ( | const tnsr::I< DataType, VolumeDim, Frame > & | normal_vector, |
| const tnsr::i< DataType, VolumeDim, Frame > & | normal_one_form | ||
| ) |
Compute projection operator onto an interface.
Returns the operator \(P^{i}_{j} = \delta^{i}_{j} - n^i n_j\), where \(n^i\) and \(n_i\) are the normal vector and normal one-form to the interface in question.
| tnsr::ii< DataType, VolumeDim, Frame > gr::transverse_projection_operator | ( | const tnsr::ii< DataType, VolumeDim, Frame > & | spatial_metric, |
| const tnsr::i< DataType, VolumeDim, Frame > & | normal_one_form | ||
| ) |
Compute projection operator onto an interface.
Returns the operator \(P_{ij} = \gamma_{ij} - n_i n_j\), where \( \gamma_{ij}\) is the spatial metric, and \( n_i\) is the normal one-form to the interface in question.