Computes the analytic data for a gauge wave solution described in [11]. 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< DataVector, 3 >, ::Tags::dt< gr::Tags::SpacetimeMetric< DataVector, 3 > >, gh::Tags::Pi< DataVector, 3 >, gh::Tags::Phi< DataVector, 3 >, gr::Tags::SpatialMetric< DataVector, 3 >, ::Tags::dt< gr::Tags::SpatialMetric< DataVector, 3 > >, Tags::Dr< gr::Tags::SpatialMetric< DataVector, 3 > >, gr::Tags::Shift< DataVector, 3 >, ::Tags::dt< gr::Tags::Shift< DataVector, 3 > >, Tags::Dr< gr::Tags::Shift< DataVector, 3 > >, 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)

	GaugeWave (double extraction_radius, double mass, double frequency, double amplitude, double peak_time, double duration)

std::unique_ptr< WorldtubeData >	get_clone () const override

void	pup (PUP::er &p) override

Public Member Functions inherited from Cce::Solutions::SphericalMetricData
	WRAPPED_PUPable_abstract (SphericalMetricData)

	SphericalMetricData (CkMigrateMessage *msg)

	SphericalMetricData (const double extraction_radius)

void	jacobian (gsl::not_null< SphericaliCartesianJ * > jacobian, size_t l_max) const

void	inverse_jacobian (gsl::not_null< CartesianiSphericalJ * > inverse_jacobian, size_t l_max) const

void	dr_inverse_jacobian (gsl::not_null< CartesianiSphericalJ * > dr_inverse_jacobian, size_t l_max) const

void	pup (PUP::er &p) override

Public Member Functions inherited from Cce::Solutions::WorldtubeData
	WRAPPED_PUPable_abstract (WorldtubeData)

	WorldtubeData (const double extraction_radius)

	WorldtubeData (CkMigrateMessage *msg)

virtual std::unique_ptr< WorldtubeData >	get_clone () const =0

template<typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const size_t output_l_max, const double time, tmpl::list< Tags... >) const
	Retrieve worldtube data represented by the analytic solution, at boundary angular resolution `l_max` and time `time` More...

void	pup (PUP::er &p) override

virtual std::unique_ptr< Cce::InitializeJ::InitializeJ< false > >	get_initialize_j (const double) const

virtual bool	use_noninertial_news () const

Static Public Attributes
static constexpr Options::String	help

Protected Member Functions
void	prepare_solution (const size_t, const double) const override
	A no-op as the gauge wave solution does not have substantial shared computation to prepare before the separate component calculations. More...

void	spherical_metric (gsl::not_null< tnsr::aa< DataVector, 3, ::Frame::Spherical<::Frame::Inertial > > * > spherical_metric, size_t l_max, double time) const 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 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 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 override
	The News vanishes, because the wave is pure gauge. 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

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

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< DataVector, 3 > >) const=0
	Computes the Cartesian spacetime metric from the spherical solution provided by the derived classes. More...

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< DataVector, 3 > > >) const=0

virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > pi, size_t output_l_max, double time, tmpl::type_< gh::Tags::Pi< DataVector, 3 > >) const

virtual void	variables_impl (gsl::not_null< tnsr::iaa< DataVector, 3 > * > d_spacetime_metric, size_t output_l_max, double time, tmpl::type_< gh::Tags::Phi< DataVector, 3 > >) const=0
	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::ii< DataVector, 3 > * > spatial_metric, size_t output_l_max, double time, tmpl::type_< gr::Tags::SpatialMetric< DataVector, 3 > >) const

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< DataVector, 3 > > >) const

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< DataVector, 3 > > >) const

virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > shift, size_t output_l_max, double time, tmpl::type_< gr::Tags::Shift< DataVector, 3 > >) const

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< DataVector, 3 > > >) const

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< DataVector, 3 > > >) const

virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > lapse, size_t output_l_max, double time, tmpl::type_< gr::Tags::Lapse< DataVector > >) const

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

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

virtual void	variables_impl (gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, -2 > > * > news, size_t output_l_max, double time, tmpl::type_< Tags::News >) const=0

void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > spacetime_metric, size_t l_max, double time, tmpl::type_< gr::Tags::SpacetimeMetric< DataVector, 3 > >) const 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< DataVector, 3 > > >) const 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_< gh::Tags::Phi< DataVector, 3 > >) const 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< DataVector, 3 > >) const 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< DataVector, 3 > > >) const 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_< gh::Tags::Phi< DataVector, 3 > >) const override
	Computes the spatial derivatives of the Cartesian spacetime metric from the spherical solution provided by the derived classes. More...

virtual void	spherical_metric (gsl::not_null< tnsr::aa< DataVector, 3, ::Frame::Spherical<::Frame::Inertial > > * > spherical_metric, size_t l_max, double time) const =0
	Must be overriden in the derived class; should compute the spacetime metric of the analytic solution in spherical coordinates. More...

virtual 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 =0
	Must be overriden in the derived class; should compute the first radial derivative of the spacetime metric of the analytic solution in spherical coordinates. More...

virtual 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 =0
	Must be overriden in the derived class; should compute the first time derivative of the spacetime metric of the analytic solution in spherical coordinates. 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

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

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< DataVector, 3 > >) const=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< DataVector, 3 > > >) const=0

virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > pi, size_t output_l_max, double time, tmpl::type_< gh::Tags::Pi< DataVector, 3 > >) const

virtual void	variables_impl (gsl::not_null< tnsr::iaa< DataVector, 3 > * > d_spacetime_metric, size_t output_l_max, double time, tmpl::type_< gh::Tags::Phi< DataVector, 3 > >) const=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< DataVector, 3 > >) const

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< DataVector, 3 > > >) const

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< DataVector, 3 > > >) const

virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > shift, size_t output_l_max, double time, tmpl::type_< gr::Tags::Shift< DataVector, 3 > >) const

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< DataVector, 3 > > >) const

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< DataVector, 3 > > >) const

virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > lapse, size_t output_l_max, double time, tmpl::type_< gr::Tags::Lapse< DataVector > >) const

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

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

virtual void	variables_impl (gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, -2 > > * > news, size_t output_l_max, double time, tmpl::type_< Tags::News >) const=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

virtual void	prepare_solution (size_t output_l_max, double time) const =0

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

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

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< DataVector, 3 > >) const =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< DataVector, 3 > > >) const =0

virtual void	variables_impl (gsl::not_null< tnsr::aa< DataVector, 3 > * > pi, size_t output_l_max, double time, tmpl::type_< gh::Tags::Pi< DataVector, 3 > >) const

virtual void	variables_impl (gsl::not_null< tnsr::iaa< DataVector, 3 > * > d_spacetime_metric, size_t output_l_max, double time, tmpl::type_< gh::Tags::Phi< DataVector, 3 > >) const =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< DataVector, 3 > >) const

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< DataVector, 3 > > >) const

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< DataVector, 3 > > >) const

virtual void	variables_impl (gsl::not_null< tnsr::I< DataVector, 3 > * > shift, size_t output_l_max, double time, tmpl::type_< gr::Tags::Shift< DataVector, 3 > >) const

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< DataVector, 3 > > >) const

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< DataVector, 3 > > >) const

virtual void	variables_impl (gsl::not_null< Scalar< DataVector > * > lapse, size_t output_l_max, double time, tmpl::type_< gr::Tags::Lapse< DataVector > >) const

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

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

virtual void	variables_impl (gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, -2 > > * > news, size_t output_l_max, double time, tmpl::type_< Tags::News >) const =0

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)

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< DataVector, 3 >, gh::Tags::Pi< DataVector, 3 >, gh::Tags::Phi< DataVector, 3 >, gr::Tags::SpatialMetric< DataVector, 3 >, gr::Tags::Shift< DataVector, 3 >, gr::Tags::Lapse< DataVector >, ::Tags::dt< gr::Tags::SpacetimeMetric< DataVector, 3 > >, ::Tags::dt< gr::Tags::SpatialMetric< DataVector, 3 > >, ::Tags::dt< gr::Tags::Shift< DataVector, 3 > >, ::Tags::dt< gr::Tags::Lapse< DataVector > >, Tags::Dr< gr::Tags::SpatialMetric< DataVector, 3 > >, Tags::Dr< gr::Tags::Shift< DataVector, 3 > >, 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 [11].

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 [11], 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

overrideprotectedvirtual

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

overrideprotectedvirtual

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.

◆ get_clone()

std::unique_ptr< WorldtubeData > Cce::Solutions::GaugeWave::get_clone ( ) const

overridevirtual

Implements Cce::Solutions::WorldtubeData.

◆ prepare_solution()

void Cce::Solutions::GaugeWave::prepare_solution	(	const	size_t,
		const double
	)		const

inlineoverrideprotectedvirtual

A no-op as the gauge wave solution does not have substantial shared computation to prepare before the separate component calculations.

Implements Cce::Solutions::WorldtubeData.

◆ 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

overrideprotectedvirtual

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 [11]):

\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() [1/20]

virtual void Cce::Solutions::WorldtubeData::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

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [2/20]

virtual void Cce::Solutions::WorldtubeData::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

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [3/20]

virtual void Cce::Solutions::WorldtubeData::variables_impl	(	gsl::not_null< Scalar< DataVector > * >	lapse,
		size_t	output_l_max,
		double	time,
		tmpl::type_< gr::Tags::Lapse< DataVector > >
	)		const

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [4/20]

void Cce::Solutions::GaugeWave::variables_impl	(	gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, -2 > > * >	News,
		size_t	l_max,
		double	time,
		tmpl::type_< Tags::News >
	)		const

overrideprotectedvirtual

The News vanishes, because the wave is pure gauge.

Implements Cce::Solutions::WorldtubeData.

◆ variables_impl() [5/20]

virtual void Cce::Solutions::WorldtubeData::variables_impl	(	gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, -2 > > * >	news,
		size_t	output_l_max,
		double	time,
		tmpl::type_< Tags::News >
	)		const

protectedvirtual

Implements Cce::Solutions::WorldtubeData.

◆ variables_impl() [6/20]

void Cce::Solutions::SphericalMetricData::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< DataVector, 3 > > >
	)		const

overrideprotected

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() [7/20]

virtual void Cce::Solutions::WorldtubeData::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< DataVector, 3 > > >
	)		const

protectedvirtual

Implements Cce::Solutions::WorldtubeData.

◆ variables_impl() [8/20]

virtual void Cce::Solutions::WorldtubeData::variables_impl	(	gsl::not_null< tnsr::aa< DataVector, 3 > * >	pi,
		size_t	output_l_max,
		double	time,
		tmpl::type_< gh::Tags::Pi< DataVector, 3 > >
	)		const

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [9/20]

void Cce::Solutions::SphericalMetricData::variables_impl	(	gsl::not_null< tnsr::aa< DataVector, 3 > * >	spacetime_metric,
		size_t	l_max,
		double	time,
		tmpl::type_< gr::Tags::SpacetimeMetric< DataVector, 3 > >
	)		const

overrideprotectedvirtual

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().

Reimplemented from Cce::Solutions::SphericalMetricData.

◆ variables_impl() [10/20]

virtual void Cce::Solutions::WorldtubeData::variables_impl	(	gsl::not_null< tnsr::aa< DataVector, 3 > * >	spacetime_metric,
		size_t	l_max,
		double	time,
		tmpl::type_< gr::Tags::SpacetimeMetric< DataVector, 3 > >
	)		const

protectedvirtual

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().

Reimplemented from Cce::Solutions::SphericalMetricData.

◆ variables_impl() [11/20]

virtual void Cce::Solutions::WorldtubeData::variables_impl	(	gsl::not_null< tnsr::i< DataVector, 3 > * >	cartesian_coordinates,
		size_t	output_l_max,
		double	time,
		tmpl::type_< Tags::CauchyCartesianCoords >
	)		const

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [12/20]

virtual void Cce::Solutions::WorldtubeData::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

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [13/20]

virtual void Cce::Solutions::WorldtubeData::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< DataVector, 3 > > >
	)		const

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [14/20]

virtual void Cce::Solutions::WorldtubeData::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< DataVector, 3 > > >
	)		const

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [15/20]

virtual void Cce::Solutions::WorldtubeData::variables_impl	(	gsl::not_null< tnsr::I< DataVector, 3 > * >	shift,
		size_t	output_l_max,
		double	time,
		tmpl::type_< gr::Tags::Shift< DataVector, 3 > >
	)		const

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [16/20]

void Cce::Solutions::SphericalMetricData::variables_impl	(	gsl::not_null< tnsr::iaa< DataVector, 3 > * >	d_spacetime_metric,
		size_t	l_max,
		double	time,
		tmpl::type_< gh::Tags::Phi< DataVector, 3 > >
	)		const

overrideprotectedvirtual

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().

Reimplemented from Cce::Solutions::SphericalMetricData.

◆ variables_impl() [17/20]

virtual void Cce::Solutions::WorldtubeData::variables_impl	(	gsl::not_null< tnsr::iaa< DataVector, 3 > * >	d_spacetime_metric,
		size_t	l_max,
		double	time,
		tmpl::type_< gh::Tags::Phi< DataVector, 3 > >
	)		const

protectedvirtual

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().

Reimplemented from Cce::Solutions::SphericalMetricData.

◆ variables_impl() [18/20]

virtual void Cce::Solutions::WorldtubeData::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< DataVector, 3 > > >
	)		const

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [19/20]

virtual void Cce::Solutions::WorldtubeData::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< DataVector, 3 > > >
	)		const

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

◆ variables_impl() [20/20]

virtual void Cce::Solutions::WorldtubeData::variables_impl	(	gsl::not_null< tnsr::ii< DataVector, 3 > * >	spatial_metric,
		size_t	output_l_max,
		double	time,
		tmpl::type_< gr::Tags::SpatialMetric< DataVector, 3 > >
	)		const

protectedvirtual

Reimplemented from Cce::Solutions::WorldtubeData.

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"}

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

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

Classes

Public Types

Public Member Functions

Static Public Attributes

Protected Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Details

Member Function Documentation

◆ dr_spherical_metric()

Details

◆ dt_spherical_metric()

Details

◆ get_clone()

◆ prepare_solution()

◆ spherical_metric()

Details

◆ variables_impl() [1/20]

◆ variables_impl() [2/20]

◆ variables_impl() [3/20]

◆ variables_impl() [4/20]

◆ variables_impl() [5/20]

◆ variables_impl() [6/20]

Details

◆ variables_impl() [7/20]

◆ variables_impl() [8/20]

◆ variables_impl() [9/20]

Details

◆ variables_impl() [10/20]

Details

◆ variables_impl() [11/20]

◆ variables_impl() [12/20]

◆ variables_impl() [13/20]

◆ variables_impl() [14/20]

◆ variables_impl() [15/20]

◆ variables_impl() [16/20]

Details

◆ variables_impl() [17/20]

Details

◆ variables_impl() [18/20]

◆ variables_impl() [19/20]

◆ variables_impl() [20/20]

Member Data Documentation

◆ help