Cce::Solutions::GaugeWave Struct Reference

Computes the analytic data for a gauge wave solution described in [7]. More...

#include <GaugeWave.hpp>

Classes
struct	Amplitude
struct	Duration
struct	ExtractionRadius
struct	Frequency
struct	Mass
struct	PeakTime

Public Types
using	options = tmpl::list< ExtractionRadius, Mass, Frequency, Amplitude, PeakTime, Duration >
Public Types inherited from Cce::Solutions::WorldtubeData
using	creatable_classes = tmpl::list< BouncingBlackHole, GaugeWave, LinearizedBondiSachs, RobinsonTrautman, RotatingSchwarzschild, TeukolskyWave >
using	tags = tmpl::list< Tags::CauchyCartesianCoords, Tags::Dr< Tags::CauchyCartesianCoords >, gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >, ::Tags::dt< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector > >, GeneralizedHarmonic::Tags::Pi< 3, ::Frame::Inertial >, GeneralizedHarmonic::Tags::Phi< 3, ::Frame::Inertial >, gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >, ::Tags::dt< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector > >, Tags::Dr< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector > >, gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >, ::Tags::dt< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector > >, Tags::Dr< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector > >, gr::Tags::Lapse< DataVector >, ::Tags::dt< gr::Tags::Lapse< DataVector > >, Tags::Dr< gr::Tags::Lapse< DataVector > >, Tags::News >
	The set of available tags provided by the analytic solution.

Public Member Functions
	WRAPPED_PUPable_decl_template (GaugeWave)
	GaugeWave (CkMigrateMessage *msg) noexcept
	GaugeWave (double extraction_radius, double mass, double frequency, double amplitude, double peak_time, double duration) noexcept
std::unique_ptr< WorldtubeData >	get_clone () const noexcept override
void	pup (PUP::er &p) noexcept override
Public Member Functions inherited from Cce::Solutions::SphericalMetricData
	WRAPPED_PUPable_abstract (SphericalMetricData)
	SphericalMetricData (CkMigrateMessage *msg) noexcept
	SphericalMetricData (const double extraction_radius) noexcept
void	jacobian (gsl::not_null< SphericaliCartesianJ * > jacobian, size_t l_max) const noexcept
void	inverse_jacobian (gsl::not_null< CartesianiSphericalJ * > inverse_jacobian, size_t l_max) const noexcept
void	dr_inverse_jacobian (gsl::not_null< CartesianiSphericalJ * > dr_inverse_jacobian, size_t l_max) const noexcept
void	pup (PUP::er &p) noexcept override
Public Member Functions inherited from Cce::Solutions::WorldtubeData
	WRAPPED_PUPable_abstract (WorldtubeData)
	WorldtubeData (const double extraction_radius) noexcept
	WorldtubeData (CkMigrateMessage *msg) noexcept
template<typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const size_t output_l_max, const double time, tmpl::list< Tags... >) const noexcept
	Retrieve worldtube data represented by the analytic solution, at boundary angular resolution l_max and time time More...
void	pup (PUP::er &p) noexcept override
virtual std::unique_ptr< Cce::InitializeJ::InitializeJ >	get_initialize_j (const double) const noexcept
virtual bool	use_noninertial_news () const noexcept

Static Public Attributes
static constexpr Options::String	help

Protected Member Functions
void	prepare_solution (const size_t, const double) const noexcept override
	A no-op as the gauge wave solution does not have substantial shared computation to prepare before the separate component calculations.
void	spherical_metric (gsl::not_null< tnsr::aa< DataVector, 3, ::Frame::Spherical<::Frame::Inertial >> * > spherical_metric, size_t l_max, double time) const noexcept override
	Compute the spherical coordinate metric from the closed-form gauge wave metric. More...
void	dr_spherical_metric (gsl::not_null< tnsr::aa< DataVector, 3, ::Frame::Spherical<::Frame::Inertial >> * > dr_spherical_metric, size_t l_max, double time) const noexcept override
	Compute the radial derivative of the spherical coordinate metric from the closed-form gauge wave metric. More...
void	dt_spherical_metric (gsl::not_null< tnsr::aa< DataVector, 3, ::Frame::Spherical<::Frame::Inertial >> * > dt_spherical_metric, size_t l_max, double time) const noexcept override
	Compute the spherical coordinate metric from the closed-form gauge wave metric. More...
void	variables_impl (gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, -2 >> * > News, size_t l_max, double time, tmpl::type_< Tags::News >) const noexcept override
	The News vanishes, because the wave is pure gauge.
virtual void	variables_impl (gsl::not_null< tnsr::i< DataVector, 3 > * > cartesian_coordinates, size_t output_l_max, double time, tmpl::type_< Tags::CauchyCartesianCoords >) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::i< DataVector, 3 > * > dr_cartesian_coordinates, size_t output_l_max, double time, tmpl::type_< Tags::Dr< Tags::CauchyCartesianCoords >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > spacetime_metric, size_t output_l_max, double time, tmpl::type_< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >>) const noexcept=0
virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > dt_spacetime_metric, size_t output_l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept=0
virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > pi, size_t output_l_max, double time, tmpl::type_< GeneralizedHarmonic::Tags::Pi< 3, ::Frame::Inertial >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::iaa< DataVector, 3 > * > d_spacetime_metric, size_t output_l_max, double time, tmpl::type_< GeneralizedHarmonic::Tags::Phi< 3, ::Frame::Inertial >>) const noexcept=0
virtual void	variables_impl (gsl::not_null< tnsr::ii< DataVector, 3 > * > spatial_metric, size_t output_l_max, double time, tmpl::type_< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::ii< DataVector, 3 > * > dt_spatial_metric, size_t output_l_max, double time, tmpl::type_< ::Tags::dt< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::ii< DataVector, 3 > * > dr_spatial_metric, size_t output_l_max, double time, tmpl::type_< Tags::Dr< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > shift, size_t output_l_max, double time, tmpl::type_< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > dt_shift, size_t output_l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > dr_shift, size_t output_l_max, double time, tmpl::type_< Tags::Dr< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > lapse, size_t output_l_max, double time, tmpl::type_< gr::Tags::Lapse< DataVector >>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > dt_lapse, size_t output_l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::Lapse< DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > dr_lapse, size_t output_l_max, double time, tmpl::type_< Tags::Dr< gr::Tags::Lapse< DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, -2 >> * > news, size_t output_l_max, double time, tmpl::type_< Tags::News >) const noexcept=0
void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > spacetime_metric, size_t l_max, double time, tmpl::type_< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >>) const noexcept override
	Computes the Cartesian spacetime metric from the spherical solution provided by the derived classes. More...
void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > dt_spacetime_metric, size_t l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept override
	Computes the time derivative of the Cartesian spacetime metric from the spherical solution provided by the derived classes. More...
void	variables_impl (gsl::not_null< tnsr::iaa< DataVector, 3 > * > d_spacetime_metric, size_t l_max, double time, tmpl::type_< GeneralizedHarmonic::Tags::Phi< 3, ::Frame::Inertial >>) const noexcept override
	Computes the spatial derivatives of the Cartesian spacetime metric from the spherical solution provided by the derived classes. More...
Protected Member Functions inherited from Cce::Solutions::SphericalMetricData
void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > spacetime_metric, size_t l_max, double time, tmpl::type_< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >>) const noexcept override
	Computes the Cartesian spacetime metric from the spherical solution provided by the derived classes. More...
void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > dt_spacetime_metric, size_t l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept override
	Computes the time derivative of the Cartesian spacetime metric from the spherical solution provided by the derived classes. More...
void	variables_impl (gsl::not_null< tnsr::iaa< DataVector, 3 > * > d_spacetime_metric, size_t l_max, double time, tmpl::type_< GeneralizedHarmonic::Tags::Phi< 3, ::Frame::Inertial >>) const noexcept override
	Computes the spatial derivatives of the Cartesian spacetime metric from the spherical solution provided by the derived classes. More...
virtual void	variables_impl (gsl::not_null< tnsr::i< DataVector, 3 > * > cartesian_coordinates, size_t output_l_max, double time, tmpl::type_< Tags::CauchyCartesianCoords >) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::i< DataVector, 3 > * > dr_cartesian_coordinates, size_t output_l_max, double time, tmpl::type_< Tags::Dr< Tags::CauchyCartesianCoords >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > spacetime_metric, size_t output_l_max, double time, tmpl::type_< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >>) const noexcept=0
virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > dt_spacetime_metric, size_t output_l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept=0
virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > pi, size_t output_l_max, double time, tmpl::type_< GeneralizedHarmonic::Tags::Pi< 3, ::Frame::Inertial >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::iaa< DataVector, 3 > * > d_spacetime_metric, size_t output_l_max, double time, tmpl::type_< GeneralizedHarmonic::Tags::Phi< 3, ::Frame::Inertial >>) const noexcept=0
virtual void	variables_impl (gsl::not_null< tnsr::ii< DataVector, 3 > * > spatial_metric, size_t output_l_max, double time, tmpl::type_< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::ii< DataVector, 3 > * > dt_spatial_metric, size_t output_l_max, double time, tmpl::type_< ::Tags::dt< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::ii< DataVector, 3 > * > dr_spatial_metric, size_t output_l_max, double time, tmpl::type_< Tags::Dr< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > shift, size_t output_l_max, double time, tmpl::type_< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > dt_shift, size_t output_l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > dr_shift, size_t output_l_max, double time, tmpl::type_< Tags::Dr< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > lapse, size_t output_l_max, double time, tmpl::type_< gr::Tags::Lapse< DataVector >>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > dt_lapse, size_t output_l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::Lapse< DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > dr_lapse, size_t output_l_max, double time, tmpl::type_< Tags::Dr< gr::Tags::Lapse< DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, -2 >> * > news, size_t output_l_max, double time, tmpl::type_< Tags::News >) const noexcept=0
Protected Member Functions inherited from Cce::Solutions::WorldtubeData
template<typename Tag >
const auto &	cache_or_compute (const size_t output_l_max, const double time) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::i< DataVector, 3 > * > cartesian_coordinates, size_t output_l_max, double time, tmpl::type_< Tags::CauchyCartesianCoords >) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::i< DataVector, 3 > * > dr_cartesian_coordinates, size_t output_l_max, double time, tmpl::type_< Tags::Dr< Tags::CauchyCartesianCoords >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > pi, size_t output_l_max, double time, tmpl::type_< GeneralizedHarmonic::Tags::Pi< 3, ::Frame::Inertial >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::ii< DataVector, 3 > * > spatial_metric, size_t output_l_max, double time, tmpl::type_< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::ii< DataVector, 3 > * > dt_spatial_metric, size_t output_l_max, double time, tmpl::type_< ::Tags::dt< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::ii< DataVector, 3 > * > dr_spatial_metric, size_t output_l_max, double time, tmpl::type_< Tags::Dr< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > shift, size_t output_l_max, double time, tmpl::type_< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > dt_shift, size_t output_l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > dr_shift, size_t output_l_max, double time, tmpl::type_< Tags::Dr< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > lapse, size_t output_l_max, double time, tmpl::type_< gr::Tags::Lapse< DataVector >>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > dt_lapse, size_t output_l_max, double time, tmpl::type_<::Tags::dt< gr::Tags::Lapse< DataVector >>>) const noexcept
virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > dr_lapse, size_t output_l_max, double time, tmpl::type_< Tags::Dr< gr::Tags::Lapse< DataVector >>>) const noexcept

Protected Attributes
double	mass_ = std::numeric_limits<double>::signaling_NaN()
double	frequency_ = std::numeric_limits<double>::signaling_NaN()
double	amplitude_ = std::numeric_limits<double>::signaling_NaN()
double	peak_time_ = std::numeric_limits<double>::signaling_NaN()
double	duration_ = std::numeric_limits<double>::signaling_NaN()
Protected Attributes inherited from Cce::Solutions::WorldtubeData
IntermediateCacheTuple	intermediate_cache_
double	extraction_radius_ = std::numeric_limits<double>::quiet_NaN()

Additional Inherited Members
Static Public Member Functions inherited from Cce::Solutions::SphericalMetricData
static void	dr_jacobian (gsl::not_null< SphericaliCartesianJ * > dr_jacobian, size_t l_max) noexcept
Protected Types inherited from Cce::Solutions::WorldtubeData
using	IntermediateCacheTuple = tuples::tagged_tuple_from_typelist< tmpl::transform< tmpl::list< Tags::CauchyCartesianCoords, Tags::Dr< Tags::CauchyCartesianCoords >, gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector >, GeneralizedHarmonic::Tags::Pi< 3, ::Frame::Inertial >, GeneralizedHarmonic::Tags::Phi< 3, ::Frame::Inertial >, gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector >, gr::Tags::Shift< 3, ::Frame::Inertial, DataVector >, gr::Tags::Lapse< DataVector >, ::Tags::dt< gr::Tags::SpacetimeMetric< 3, ::Frame::Inertial, DataVector > >, ::Tags::dt< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector > >, ::Tags::dt< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector > >, ::Tags::dt< gr::Tags::Lapse< DataVector > >, Tags::Dr< gr::Tags::SpatialMetric< 3, ::Frame::Inertial, DataVector > >, Tags::Dr< gr::Tags::Shift< 3, ::Frame::Inertial, DataVector > >, Tags::Dr< gr::Tags::Lapse< DataVector > >, Tags::News >, tmpl::bind< IntermediateCacheTag, tmpl::_1 > >>

Detailed Description

Computes the analytic data for a gauge wave solution described in [7].

Details

This test computes an analytic solution of a pure-gauge perturbation of the Schwarzschild metric. The gauge perturbation is constructed using the time-dependent coordinate transformation of the ingoing Eddington-Finklestein coordinate \(\nu \rightarrow \nu + F(t - r) / r\), where

\[ F(u) = A \sin(\omega u) e^{- (u - u_0)^2 / k^2}. \]

Note

In the paper [7], a translation map was applied to the solution to make the test more demanding. For simplicity, we omit that extra map. The behavior of translation-independence is tested by the Cce::Solutions::BouncingBlackHole solution.

Member Function Documentation

dr_spherical_metric()

void Cce::Solutions::GaugeWave::dr_spherical_metric ( gsl::not_null< tnsr::aa< DataVector, 3, ::Frame::Spherical<::Frame::Inertial >> * >  dr_spherical_metric, size_t  l_max, double  time  ) const

overrideprotectedvirtualnoexcept

Compute the radial derivative of the spherical coordinate metric from the closed-form gauge wave metric.

Details

The transformation of the ingoing Eddington-Finkelstein coordinate produces the radial derivative of the metric components in spherical coordinates:

\begin{align*} \partial_r g_{tt} + \partial_t g_{tt} =& \frac{2}{r^4} \left[r + \partial_u F(u)\right] \left[- M r + (r - 3 M)\partial_u F(u)\right] \\ \partial_r g_{rt} + \partial_t g_{rt} =& - \frac{1}{r^5} \left\{2 M r^3 + 2 r F(u) (r - 3M) + \partial_u F(u)[r^3 + F(u)(3r - 8 M)] + 2 r [\partial_u F(u)]^2 (r - 3M)\right\}\\ \partial_r g_{rr} + \partial_t g_{rr} =& \frac{2}{r^6} \left\{- M r^4 + F(u)^2 (2r - 5M) + \partial_u F(u) r^2 \left[4 M r + \partial_u F(u) (r - 3 M)\right] + F(u) r \left[6 M r + \partial_u F(u) (3r - 8 M)\right]\right\} \\ g_{\theta \theta} =& 2 r \\ g_{\phi \phi} =& 2 r \sin^2(\theta), \end{align*}

and all other components vanish (these formulae are obtained simply by applying radial derivatives to those given in GaugeWave::spherical_metric()). Here, \(F(u)\) is defined as

\begin{align*} F(u) &= A \sin(\omega u) e^{-(u - u_0)^2 /k^2},\\ \partial_u F(u) &= A \left[-2 \frac{u - u_0}{k^2} \sin(\omega u) + \omega \cos(\omega u)\right] e^{-(u - u_0)^2 / k^2}. \end{align*}

Warning

The \(\phi\) components are returned in a form for which the \(\sin(\theta)\) factors are omitted, assuming that derivatives and Jacobians will be applied similarly omitting those factors (and therefore improving precision of the tensor expression). If you require the \(\sin(\theta)\) factors, be sure to put them in by hand in the calling code.

Implements Cce::Solutions::SphericalMetricData.

dt_spherical_metric()

void Cce::Solutions::GaugeWave::dt_spherical_metric ( gsl::not_null< tnsr::aa< DataVector, 3, ::Frame::Spherical<::Frame::Inertial >> * >  dt_spherical_metric, size_t  l_max, double  time  ) const

overrideprotectedvirtualnoexcept

Compute the spherical coordinate metric from the closed-form gauge wave metric.

Details

The transformation of the ingoing Eddington-Finkelstein coordinate produces metric components in spherical coordinates:

\begin{align*} \partial_t g_{tt} =& \frac{-2 \partial_u^2 F(u)}{r^3} \left(r - 2 M\right) \left(r + \partial_u F(u)\right) \\ \partial_t g_{rt} =& \frac{1}{r^4} \Bigg\{\partial_u^2 F(u) \left[2 M r^2 + \left(r - 2 M\right) (r \partial_u F(u) + F(u))\right] \\ &+ \left[r + \partial_u F(u)\right] \left(r - 2M\right) \left[r \partial_u^2 F(u) + \partial_u F(u) \right]\Bigg\} \\ \partial_t g_{rr} =& \frac{1}{r^5}\Bigg\{-\left[r \partial_u^2 F(u) + \partial_u F(u)\right] \left[r^3 + 2 M r^2 + \left(r - 2 M\right) \left(r \partial_u F(u) + F(u)\right)\right]\\ &+ \left[r^2 - r \partial_u F(u) - F(u)\right] \left(r - 2 M\right) \left[r \partial_u^2 F(u) + \partial_u F(u)\right]\Bigg\} \\ \partial_t g_{\theta \theta} =& 0 \\ \partial_t g_{\phi \phi} =& 0, \end{align*}

and all other components vanish. Here, \(F(u)\) is defined as

\begin{align*} F(u) &= A \sin(\omega u) e^{-(u - u_0)^2 /k^2},\\ \partial_u F(u) &= A \left[-2 \frac{u - u_0}{k^2} \sin(\omega u) + \omega \cos(\omega u)\right] e^{-(u - u_0)^2 / k^2},\\ \partial^2_u F(u) &= \frac{A}{k^4} \left\{-4 k^2 \omega (u - u_0) \cos(\omega u) + \left[-2 k^2 + 4 (u - u_0)^2 - k^4 \omega^2\right] \sin(\omega u)\right\} e^{-(u - u_0) / k^2} \end{align*}

Warning

The \(\phi\) components are returned in a form for which the \(\sin(\theta)\) factors are omitted, assuming that derivatives and Jacobians will be applied similarly omitting those factors (and therefore improving precision of the tensor expression). If you require the \(\sin(\theta)\) factors, be sure to put them in by hand in the calling code.

Implements Cce::Solutions::SphericalMetricData.

spherical_metric()

void Cce::Solutions::GaugeWave::spherical_metric ( gsl::not_null< tnsr::aa< DataVector, 3, ::Frame::Spherical<::Frame::Inertial >> * >  spherical_metric, size_t  l_max, double  time  ) const

overrideprotectedvirtualnoexcept

Compute the spherical coordinate metric from the closed-form gauge wave metric.

Details

The transformation of the ingoing Eddington-Finkelstein coordinate produces metric components in spherical coordinates (identical up to minor manipulations of the metric given in Eq. (149) of [7]):

\begin{align*} g_{tt} &= \frac{-1}{r^3}\left(r - 2 M\right) \left[r + \partial_u F(u)\right]^2\\ g_{rt} &= \frac{1}{r^4} \left[r + \partial_u F(u)\right] \left\{2 M r^2 + \left(r - 2 M\right) \left[r \partial_u F(u) + F(u)\right]\right\} \\ g_{rr} &= \frac{1}{r^5} \left[r^2 - r \partial_u F(u) - F(u)\right] \left\{r^3 + 2 M r^2 + \left(r - 2 M\right) \left[r \partial_u F(u) + F(u)\right]\right\} \\ g_{\theta \theta} &= r^2 \\ g_{\phi \phi} &= r^2 \sin^2(\theta), \end{align*}

and all other components vanish. Here, \(F(u)\) is defined as

\begin{align*} F(u) &= A \sin(\omega u) e^{-(u - u_0)^2 /k^2},\\ \partial_u F(u) &= A \left[-2 \frac{u - u_0}{k^2} \sin(\omega u) + \omega \cos(\omega u)\right] e^{-(u - u_0)^2 / k^2}. \end{align*}

Warning

The \(\phi\) components are returned in a form for which the \(\sin(\theta)\) factors are omitted, assuming that derivatives and Jacobians will be applied similarly omitting those factors (and therefore improving precision of the tensor expression). If you require the \(\sin(\theta)\) factors, be sure to put them in by hand in the calling code.

Implements Cce::Solutions::SphericalMetricData.

variables_impl() [1/3]

void Cce::Solutions::SphericalMetricData::variables_impl

overrideprotectednoexcept

Computes the time derivative of the Cartesian spacetime metric from the spherical solution provided by the derived classes.

Details

The derived classes provide the time derivative of the spherical metric data via the virtual function SphericalMetricData::dt_spherical_metric() at a resolution determined by the l_max argument. This function performs the coordinate transformation using the Jacobian computed from SphericalMetricData::inverse_jacobian().

variables_impl() [2/3]

void Cce::Solutions::SphericalMetricData::variables_impl

overrideprotectednoexcept

Computes the Cartesian spacetime metric from the spherical solution provided by the derived classes.

Details

The derived classes provide spherical metric data via the virtual function SphericalMetricData::spherical_metric() at a resolution determined by the l_max argument. This function performs the coordinate transformation using the Jacobian computed from SphericalMetricData::inverse_jacobian().

variables_impl() [3/3]

void Cce::Solutions::SphericalMetricData::variables_impl

overrideprotectednoexcept

Computes the spatial derivatives of the Cartesian spacetime metric from the spherical solution provided by the derived classes.

Details

The derived classes provide the radial derivative of the spherical metric data via the virtual function SphericalMetricData::dr_spherical_metric() at a resolution determined by the l_max_ argument. This function performs the additional angular derivatives necessary to assemble the full spatial derivative and performs the coordinate transformation to Cartesian coordinates via the Jacobians computed in SphericalMetricData::inverse_jacobian() and SphericalMetricData::inverse_jacobian().

Member Data Documentation

help

constexpr Options::String Cce::Solutions::GaugeWave::help

staticconstexpr

Initial value:

= {
      "Analytic solution representing worldtube data for a pure-gauge "
      "perturbation near a Schwarzschild metric in spherical coordinates"}

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The documentation for this struct was generated from the following file:

• src/Evolution/Systems/Cce/AnalyticSolutions/GaugeWave.hpp