Computes the analytic data for a Teukolsky wave solution described in [11]. More...
#include <TeukolskyWave.hpp>
Classes | |
| struct | Amplitude |
| struct | Duration |
| struct | ExtractionRadius |
Public Member Functions | |
| WRAPPED_PUPable_decl_template (TeukolskyWave) | |
| TeukolskyWave (CkMigrateMessage *msg) | |
| TeukolskyWave (double extraction_radius, double amplitude, 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 Teukolsky 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 perturbative Teukolsky 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 perturbative Teukolsky 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 time derivative of the spherical coordinate metric from the closed-form perturbative Teukolsky 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 |
| Compute the news associated with the (2,0)-mode Teukolsky wave. 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... | |
| 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 |
| 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 | amplitude_ = 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() |
Detailed Description
Computes the analytic data for a Teukolsky wave solution described in [11].
Details
This test computes an outgoing perturbative wave solution in spherical coordinates with wave profile
\[ F(u) = A e^{- u^2 / k^2}. \]
Member Function Documentation
◆ dr_spherical_metric()
|
overrideprotectedvirtual |
Compute the radial derivative of the spherical coordinate metric from the closed-form perturbative Teukolsky wave metric.
Details
The specific outgoing wave selected in this analytic solution is constructed from a (2,0) mode as in [11], and takes the form
\begin{align*} \partial_r g_{rr} &= f_{r r} \partial_r A \\ \partial_r g_{r \theta} &= f_{r \theta} (B + r \partial_r B)\\ \partial_r g_{\theta \theta} &= 2 (1 + C f_{\theta \theta}^{(C)} + A f_{\theta \theta}^{(A)}) r + (\partial_r C f_{\theta \theta}^{(C)} + \partial_r A f_{\theta \theta}^{(A)}) r^2 \\ \partial_r g_{\phi \phi} &= 2 (1 + C f_{\phi \phi}^{(C)} + A f_{\phi \phi}^{(A)}) r \sin^2 \theta + (\partial_r C f_{\phi \phi}^{(C)} + \partial_r A f_{\phi \phi}^{(A)}) r^2 \sin^2 \theta\\ \end{align*}
and all other components vanish. The angular factors \(f_{a b}\) and the metric component functions \(A, B,\) and \(C\) are defined as in TeukolskyWave::spherical_metric(). The radial derivatives of the pulse profile functions are obtained by:
\begin{align*} \partial_r A + \partial_t A &= \frac{-9 a e^{-u^2/k^2}}{k^4 r^6} \left( 5 k^4 + 4 r^2 u^2 - 2 k^2 r (r + 4 u)\right), \\ \partial_r B + \partial_t B &= \frac{2 a e^{-u^2/k^2}}{k^6 r^6} \left( 15 k^6 - 8 r^3 u^3 + 6 k^2 r^2 u (2 r + 3 u) - 3 k^4 r (3 r + 8 u)\right), \\ \partial_r C + \partial_t C &= \frac{-a e^{-u^2/k^2}}{4 k^8 r^6} \left( 105 k^8 + 16 k^4 u^4 - 16 k^2 r^3 u^2 (3 r + 2 u) - 6 k^6 r (9 r + 28 u) + 12 k^4 r^2 (r^2 + 4 r u + 9 u^2)\right), \end{align*}
and the time derivatives of the pulse profile functions are given in TeukolskyWave::dt_spherical_metric().
- Note
- 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()
|
overrideprotectedvirtual |
Compute the time derivative of the spherical coordinate metric from the closed-form perturbative Teukolsky wave metric.
Details
The specific outgoing wave selected in this analytic solution is constructed from a (2,0) mode as in [11], and takes the form
\begin{align*} \partial_t g_{rr} &= f_{r r} \partial_t A \\ \partial_t g_{r \theta} &= f_{r \theta} r \partial_t B\\ \partial_t g_{\theta \theta} &= (\partial_t C f_{\theta \theta}^{(C)} + \partial_t A f_{\theta \theta}^{(A)}) r^2 \\ \partial_t g_{\phi \phi} &= (\partial_t C f_{\phi \phi}^{(C)} + \partial_t A f_{\phi \phi}^{(A)}) r^2 \sin^2 \theta\\ \end{align*}
and all other components vanish. The angular factors \(f_{a b}\) and the metric component functions \(A, B,\) and \(C\) are defined as in TeukolskyWave::spherical_metric(). The time derivatives of the pulse profile functions are:
\begin{align*} \partial_t A &= \frac{-2 u}{k^2} A + \frac{3 a e^{-u^2/k^2}}{k^4 r^5} \left( 8 r^2 u - 6 k^2 r \right), \\ \partial_t B &= \frac{-2 u}{k^2} B + \frac{2 a e^{-u^2/k^2}}{k^6 r^5} \left(12 r^3 u^2 - 6 k^2 r^2 (r + 2 u) + 6 k^4 r\right), \\ \partial_t C &= \frac{-2 u}{k^2} C + \frac{-a e^{-u^2/k^2}}{4 k^8 r^5} \left(64 k^4 u^3 - 16 k^2 r^3 u (6 r + 3 u) - 42 k^6 r + 12 k^4 r^2 (2 r + 6 u)\right), \end{align*}
- Note
- 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()
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overridevirtual |
Implements Cce::Solutions::WorldtubeData.
◆ prepare_solution()
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inlineoverrideprotectedvirtual |
A no-op as the Teukolsky wave solution does not have substantial shared computation to prepare before the separate component calculations.
Implements Cce::Solutions::WorldtubeData.
◆ spherical_metric()
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overrideprotectedvirtual |
Compute the spherical coordinate metric from the closed-form perturbative Teukolsky wave metric.
Details
The specific outgoing wave selected in this analytic solution is constructed from a (2,0) mode as in [11], and takes the form
\begin{align*} g_{tt} &= -1\\ g_{rr} &= (1 + A f_{rr}) \\ g_{r \theta} &= 2 B f_{r \theta} r\\ g_{\theta \theta} &= (1 + C f_{\theta \theta}^{(C)} + A f_{\theta \theta}^{(A)}) r^2\\ g_{\phi \phi} &= (1 + C f_{\phi \phi}^{(C)} + A f_{\phi \phi}^{(A)}) r^2 \sin^2 \theta\\ \end{align*}
and all other components vanish. The angular factors generated by the choice of spin-weighted spherical harmonic are
\begin{align*} f_{rr} &= 2 - 3 \sin^2 \theta \\ f_{r \theta} &= -3 \sin \theta \cos \theta \\ f_{\theta \theta}^{(C)} &= 3 \sin^2 \theta \\ f_{\theta \theta}^{(A)} &= -1 \\ f_{\phi \phi}^{(C)} &= - 3 \sin^2 \theta \\ f_{\phi \phi}^{(A)} &= 3 \sin^2 \theta -1, \end{align*}
the radial and time dependent factors are
\begin{align*} A &= 3 \left(\frac{\partial_u^2 F(u)}{r^3} + \frac{3 \partial_u F(u)}{r^4} + \frac{3 F(u)}{r^5} \right),\\ B &= - \left(\frac{\partial_u^3 F(u)}{r^2} + \frac{3 \partial_u^2 F(u)}{r^3} + \frac{6 \partial_uF(u)}{r^4} + \frac{6 F(u)}{r^5}\right), \\ C &= \frac{1}{4} \left(\frac{\partial_u^4 F(u)}{r} + \frac{2 \partial_u^3 F(u)}{r^2} + \frac{9 \partial_u^2 F(u)}{r^3} + \frac{21 \partial_u F(u)}{r^4} + \frac{21 F(u)}{r}\right), \end{align*}
and the pulse profile is
\[ F(u) = a e^{-u^2 /k^2}. \]
So, the pulse profile factors expand to
\begin{align*} A &= \frac{3 a e^{-u^2/k^2}}{k^4 r^5} \left(3 k^4 + 4 r^2 u^2 - 2 k^2 r (r + 3 u)\right),\\ B &= \frac{2 a e^{-u^2/k^2}}{k^6 r^5} \left(-3 k^6 + 4 r^3 u^3 - 6 k^2 r^2 u (r + u) + 3 k^4 r (r + 2 u)\right), \\ C &= \frac{a e^{-u^2/k^2}}{4 k^8 r^5} \left(21 k^8 + 16 r^4 u^4 - 16 k^2 r^3 u^2 (3 r + u) - 6 k^6 r (3 r + 7 u) + 12 k^4 r^2 (r^2 + 2 r u + 3 u^2)\right), \end{align*}
- Note
- 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() [1/36]
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Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [2/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [3/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [4/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [5/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [6/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [7/36]
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overrideprotectedvirtual |
Compute the news associated with the (2,0)-mode Teukolsky wave.
Details
The value of the news is
\begin{align*} N = \frac{3 \sin^2 \theta}{4} \partial_u^5 F(u) \end{align*}
where \(F(u)\) is the pulse profile, taken to be
\[ F(u) = a e^{-u^2 /k^2}, \]
So, the news expands to
\[ N = -\frac{6 a e^{-u^2/k^2} u}{k^{10}} \left(15 k^4 - 20 k^2 u^2 + 4 u^4\right) \]
in this analytic solution.
Implements Cce::Solutions::WorldtubeData.
◆ variables_impl() [8/36]
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Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [9/36]
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Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [10/36]
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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() [11/36]
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Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [12/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [13/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [14/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [15/36]
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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() [16/36]
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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() [17/36]
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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() [18/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [19/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [20/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [21/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [22/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [23/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [24/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [25/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [26/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [27/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [28/36]
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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() [29/36]
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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() [30/36]
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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() [31/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [32/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [33/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [34/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [35/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
◆ variables_impl() [36/36]
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protectedvirtual |
Reimplemented from Cce::Solutions::SphericalMetricData.
Member Data Documentation
◆ help
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staticconstexpr |
The documentation for this struct was generated from the following file:
- src/Evolution/Systems/Cce/AnalyticSolutions/TeukolskyWave.hpp