gr Namespace Reference

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 Dim, typename Frame >
void	geodesic_equation (gsl::not_null< tnsr::I< DataType, Dim, Frame > * > dt_x, gsl::not_null< tnsr::i< DataType, Dim, Frame > * > dt_pi, gsl::not_null< Scalar< DataType > * > dt_lnp0, const tnsr::I< DataType, Dim, Frame > &x, const tnsr::i< DataType, Dim, Frame > &pi, const Scalar< DataType > &lnp0, const Scalar< DataType > &lapse, const tnsr::i< DataType, Dim, Frame > &deriv_lapse, const tnsr::I< DataType, Dim, Frame > &shift, const tnsr::iJ< DataType, Dim, Frame > &deriv_shift, const tnsr::II< DataType, Dim, Frame > &inv_spatial_metric, const tnsr::iJJ< DataType, Dim, Frame > &deriv_inv_spatial_metric, const tnsr::ii< DataType, Dim, Frame > &extrinsic_curvature)
	First-order formulation of the geodesic equation for null geodesics that is suitable for ray tracing. 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_vector, const tnsr::I< DataType, VolumeDim, Frame > &interface_unit_normal_vector, 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<typename DataType >
DataType	tortoise_radius_from_boyer_lindquist_minus_r_plus (const DataType &r_minus_r_plus, double mass, double dimensionless_spin)
	Computes the tortoise coordinates radius from the Boyer-Lindquist radius. More...
template<typename DataType >
DataType	boyer_lindquist_radius_minus_r_plus_from_tortoise (const DataType &r_star, double mass, double dimensionless_spin)
	Computes the Boyer-Lindquist radius from tortoise coordinates. 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 (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 (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 SpatialDim, typename Frame >
tnsr::ijj< DataType, SpatialDim, Frame >	covariant_derivative_of_extrinsic_curvature (const tnsr::ijj< DataType, SpatialDim, Frame > &d_ex_curv, const tnsr::ii< DataType, SpatialDim, Frame > &ex_curv, const tnsr::Ijj< DataType, SpatialDim, Frame > &spatial_christoffel_second_kind)
	Computes the spatial covariant derivative of the extrinsic curvature. More...
template<typename DataType , size_t SpatialDim, typename Frame >
void	covariant_derivative_of_extrinsic_curvature (gsl::not_null< tnsr::ijj< DataType, SpatialDim, Frame > * > grad_ex_curv, const tnsr::ijj< DataType, SpatialDim, Frame > &d_ex_curv, const tnsr::ii< DataType, SpatialDim, Frame > &ex_curv, const tnsr::Ijj< DataType, SpatialDim, Frame > &spatial_christoffel_second_kind)
	Computes the spatial covariant derivative of the extrinsic curvature. 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, 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, 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 (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 (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<typename Frame >
void	pontryagin_scalar_in_vacuum (gsl::not_null< Scalar< DataVector > * > pontryagin_scalar, const tnsr::ii< DataVector, 3, Frame > &weyl_electric, const tnsr::ii< DataVector, 3, Frame > &weyl_magnetic, const tnsr::II< DataVector, 3, Frame > &inverse_spatial_metric)
	Computes the Pontryagin scalar in vacuum. More...
template<typename Frame >
Scalar< DataVector >	pontryagin_scalar_in_vacuum (const tnsr::ii< DataVector, 3, Frame > &weyl_electric, const tnsr::ii< DataVector, 3, Frame > &weyl_magnetic, const tnsr::II< DataVector, 3, Frame > &inverse_spatial_metric)
	Computes the Pontryagin scalar in vacuum. More...
void	gauss_bonnet_scalar_in_vacuum (gsl::not_null< Scalar< DataVector > * > gb_scalar, const Scalar< DataVector > &weyl_electric_scalar, const Scalar< DataVector > &weyl_magnetic_scalar)
	Computes Gauss-Bonnet scalar in vacuum. More...
Scalar< DataVector >	gauss_bonnet_scalar_in_vacuum (const Scalar< DataVector > &weyl_electric_scalar, const Scalar< DataVector > &weyl_magnetic_scalar)
	Computes Gauss-Bonnet scalar in vacuum. More...
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 (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, 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, 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 (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 (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...

Detailed Description

Holds functions related to general relativity.

Function Documentation

boyer_lindquist_radius_minus_r_plus_from_tortoise()

template<typename DataType >

DataType gr::boyer_lindquist_radius_minus_r_plus_from_tortoise	(	const DataType &	r_star,
		double	mass,
		double	dimensionless_spin
	)

Computes the Boyer-Lindquist radius from tortoise coordinates.

This function inverts the transformation from tortoise coordinates radius \(r_*\) to Boyer-Lindquist radius \(r\):

\begin{equation} r_* = r + \frac{2 M}{r_+ - r_-}\left( r_+ \ln(\frac{r - r_+}{2 M}) - r_- \ln(\frac{r - r_-}{2 M}) \right) \end{equation}

where \(r_\pm = M \pm \sqrt{M^2 - a^2}\).

It performs a numerical rootfind to invert the above equation.

Parameters

r_star	Tortoise coordinate \(r_*\).
mass	Kerr mass parameter \(M\).
dimensionless_spin	Kerr dimensionless spin parameter \(\chi=a/M\).

Returns: Boyer-Lindquist radius minus \(r_+\): \(r - r_+\).

gauss_bonnet_scalar_in_vacuum() [1/2]

Scalar< DataVector > gr::gauss_bonnet_scalar_in_vacuum	(	const Scalar< DataVector > &	weyl_electric_scalar,
		const Scalar< DataVector > &	weyl_magnetic_scalar
	)

Computes Gauss-Bonnet scalar in vacuum.

Details

The Gauss-Bonnet scalar in vacuum is given by

\begin{align} \mathcal{G} &\equiv R_{abcd} R^{abcd} - 4 R_{ab} R^{ab} + R^2 &= C_{abcd} C^{abcd} &= 8 (E_{ab} E^{ab} - B_{ab} B^{ab}) ~, \end{align}

where \( R_{abcd} \), \( R_{ab} \), \( R \) \( C_{abcd} \) are the Riemann tensor, Ricci tensor, Ricci scalar and Weyl tensor in 4 spacetime dimensions. The Gauss-Bonnet scalar in vacuum can be computed in terms of the electric ( \( E_{ab} \)) and magnetic ( \( B_{ab} \)) parts of the Weyl tensor.

See also

gr::Tags::WeylMagnetic and gr::Tags::WeylElectric

gauss_bonnet_scalar_in_vacuum() [2/2]

void gr::gauss_bonnet_scalar_in_vacuum	(	gsl::not_null< Scalar< DataVector > * >	gb_scalar,
		const Scalar< DataVector > &	weyl_electric_scalar,
		const Scalar< DataVector > &	weyl_magnetic_scalar
	)

Computes Gauss-Bonnet scalar in vacuum.

Details

The Gauss-Bonnet scalar in vacuum is given by

\begin{align} \mathcal{G} &\equiv R_{abcd} R^{abcd} - 4 R_{ab} R^{ab} + R^2 &= C_{abcd} C^{abcd} &= 8 (E_{ab} E^{ab} - B_{ab} B^{ab}) ~, \end{align}

where \( R_{abcd} \), \( R_{ab} \), \( R \) \( C_{abcd} \) are the Riemann tensor, Ricci tensor, Ricci scalar and Weyl tensor in 4 spacetime dimensions. The Gauss-Bonnet scalar in vacuum can be computed in terms of the electric ( \( E_{ab} \)) and magnetic ( \( B_{ab} \)) parts of the Weyl tensor.

See also

gr::Tags::WeylMagnetic and gr::Tags::WeylElectric

geodesic_acceleration() [1/2]

template<typename DataType , size_t Dim>

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.

Details

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.

geodesic_acceleration() [2/2]

template<typename DataType , size_t Dim>

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.

Details

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.

geodesic_equation()

template<typename DataType , size_t Dim, typename Frame >

void gr::geodesic_equation	(	gsl::not_null< tnsr::I< DataType, Dim, Frame > * >	dt_x,
		gsl::not_null< tnsr::i< DataType, Dim, Frame > * >	dt_pi,
		gsl::not_null< Scalar< DataType > * >	dt_lnp0,
		const tnsr::I< DataType, Dim, Frame > &	x,
		const tnsr::i< DataType, Dim, Frame > &	pi,
		const Scalar< DataType > &	lnp0,
		const Scalar< DataType > &	lapse,
		const tnsr::i< DataType, Dim, Frame > &	deriv_lapse,
		const tnsr::I< DataType, Dim, Frame > &	shift,
		const tnsr::iJ< DataType, Dim, Frame > &	deriv_shift,
		const tnsr::II< DataType, Dim, Frame > &	inv_spatial_metric,
		const tnsr::iJJ< DataType, Dim, Frame > &	deriv_inv_spatial_metric,
		const tnsr::ii< DataType, Dim, Frame > &	extrinsic_curvature
	)

First-order formulation of the geodesic equation for null geodesics that is suitable for ray tracing.

This is the formulation of the geodesic equation that is originally used for ray tracing in [25] (Eq. (4)) and [26] (Eq. (6)).

For null rays with position \(x^\mu\), proper time (or affine geodesic parameter) \(\tau\), and four-momentum \(p^\mu = dx^\mu / d\tau\), we first define the momentum variable

\begin{equation} \Pi_i = \frac{p_i}{\alpha p^0} = \frac{p_i}{\sqrt{\gamma^{jk} p_j p_k}} \text{.} \end{equation}

Here we work in a 3+1 decomposition of spacetime with time coordinate \(t\), lapse \(\alpha\), shift \(\beta^i\), spatial metric \(\gamma_{ij}\), and extrinsic curvature \(K_{ij}\). Then, the geodesic equation is given by

\begin{align} \frac{d \Pi_i}{d t} &= -\partial_i \alpha + (\Pi^j \partial_j \alpha -\alpha K_{jk}\Pi^j\Pi^k) \Pi_i) + \Pi_k \partial_i \beta^k - \frac{1}{2} \alpha \Pi_j\Pi_k \partial_i \gamma^{jk} \\ \frac{d x^i}{d t} &= \alpha \Pi^i - \beta^i \text{.} \end{align}

This function also computes the evolution of the additional redshift variable \(\ln(\alpha p^0)\) (Eq. (5) in [25]) as

\begin{equation} \frac{d \ln(\alpha p^0)}{d t} = -\Pi^i \partial_i \alpha + \alpha K_{ij} \Pi^i \Pi^j \text{.} \end{equation}

interface_null_normal()

template<typename DataType , size_t VolumeDim, typename Frame >

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,
		double	sign
	)

Compute null normal vector to the boundary of a closed region in a spatial slice of spacetime.

Details

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*}

pontryagin_scalar_in_vacuum() [1/2]

template<typename Frame >

Scalar< DataVector > gr::pontryagin_scalar_in_vacuum	(	const tnsr::ii< DataVector, 3, Frame > &	weyl_electric,
		const tnsr::ii< DataVector, 3, Frame > &	weyl_magnetic,
		const tnsr::II< DataVector, 3, Frame > &	inverse_spatial_metric
	)

Computes the Pontryagin scalar in vacuum.

Details

The Pontryagin scalar in vacuum is given by

\begin{align} \mathcal{P} &\equiv {^{\star} C}_{abcd} C^{abcd} \\ &= - 16 E_{ab} B^{ab} ~, \end{align}

where \( C_{abcd} \) it the Weyl tensor (with dual \( {^\star} C}_{abcd} \)) in 4 spacetime dimensions. Here it is computed in terms of the electric ( \( E_{ab} \)) and magnetic ( \( B_{ab} \)) parts of the Weyl scalar, with the conventions used here for ( \( \{E_{ab}, B_{ab}\} \)).

See also

gr::Tags::WeylMagnetic and gr::Tags::WeylElectric

pontryagin_scalar_in_vacuum() [2/2]

template<typename Frame >

void gr::pontryagin_scalar_in_vacuum	(	gsl::not_null< Scalar< DataVector > * >	pontryagin_scalar,
		const tnsr::ii< DataVector, 3, Frame > &	weyl_electric,
		const tnsr::ii< DataVector, 3, Frame > &	weyl_magnetic,
		const tnsr::II< DataVector, 3, Frame > &	inverse_spatial_metric
	)

Computes the Pontryagin scalar in vacuum.

Details

The Pontryagin scalar in vacuum is given by

\begin{align} \mathcal{P} &\equiv {^{\star} C}_{abcd} C^{abcd} \\ &= - 16 E_{ab} B^{ab} ~, \end{align}

where \( C_{abcd} \) it the Weyl tensor (with dual \( {^\star} C}_{abcd} \)) in 4 spacetime dimensions. Here it is computed in terms of the electric ( \( E_{ab} \)) and magnetic ( \( B_{ab} \)) parts of the Weyl scalar, with the conventions used here for ( \( \{E_{ab}, B_{ab}\} \)).

See also

gr::Tags::WeylMagnetic and gr::Tags::WeylElectric

spacetime_normal_one_form() [1/2]

template<typename DataType , size_t SpatialDim, typename Frame >

tnsr::a< DataType, SpatialDim, Frame > gr::spacetime_normal_one_form

(

const Scalar< DataType > &

lapse

)

Computes spacetime normal one-form from lapse.

Details

If \(\alpha\) is the lapse, then

\begin{align} n_t &= - \alpha \\ n_i &= 0 \end{align}

is computed.

spacetime_normal_one_form() [2/2]

template<typename DataType , size_t SpatialDim, typename Frame >

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.

Details

If \(\alpha\) is the lapse, then

\begin{align} n_t &= - \alpha \\ n_i &= 0 \end{align}

is computed.

tortoise_radius_from_boyer_lindquist_minus_r_plus()

template<typename DataType >

DataType gr::tortoise_radius_from_boyer_lindquist_minus_r_plus	(	const DataType &	r_minus_r_plus,
		double	mass,
		double	dimensionless_spin
	)

Computes the tortoise coordinates radius from the Boyer-Lindquist radius.

This function evaluates the transformation from tortoise coordinates \(r_*\) to Boyer-Lindquist radius \(r\):

\begin{equation} r_* = r + \frac{2 M}{r_+ - r_-}\left( r_+ \ln(\frac{r - r_+}{2 M}) - r_- \ln(\frac{r - r_-}{2 M}) \right) \end{equation}

where \(r_\pm = M \pm \sqrt{M^2 - a^2}\).

Parameters

r_minus_r_plus	Boyer-Lindquist radius minus \(r_+\): \(r - r_+\).
mass	Kerr mass parameter \(M\).
dimensionless_spin	Kerr dimensionless spin parameter \(\chi=a/M\).

Returns: Tortoise coordinates radius \(r_*\).

transverse_projection_operator() [1/3]

template<typename DataType , size_t VolumeDim, typename Frame >

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.

Details

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*}

transverse_projection_operator() [2/3]

template<typename DataType , size_t VolumeDim, typename Frame >

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.

Details

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.

transverse_projection_operator() [3/3]

template<typename DataType , size_t VolumeDim, typename Frame >

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.

Details

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.