Gauge wave in flat spacetime. More...
#include <GaugeWave.hpp>
Classes | |
| struct | Amplitude |
| struct | Wavelength |
Public Types | |
| using | options = tmpl::list< Amplitude, Wavelength > |
| template<typename DataType > | |
| using | DerivLapse = ::Tags::deriv< gr::Tags::Lapse< DataType >, tmpl::size_t< volume_dim >, Frame::Inertial > |
| template<typename DataType > | |
| using | DerivShift = ::Tags::deriv< gr::Tags::Shift< DataType, volume_dim >, tmpl::size_t< volume_dim >, Frame::Inertial > |
| template<typename DataType > | |
| using | DerivSpatialMetric = ::Tags::deriv< gr::Tags::SpatialMetric< DataType, volume_dim >, tmpl::size_t< volume_dim >, Frame::Inertial > |
| template<typename DataType > | |
| using | tags = tmpl::list< gr::Tags::Lapse< DataType >, ::Tags::dt< gr::Tags::Lapse< DataType > >, DerivLapse< DataType >, gr::Tags::Shift< DataType, volume_dim >, ::Tags::dt< gr::Tags::Shift< DataType, volume_dim > >, DerivShift< DataType >, gr::Tags::SpatialMetric< DataType, volume_dim >, ::Tags::dt< gr::Tags::SpatialMetric< DataType, volume_dim > >, DerivSpatialMetric< DataType >, gr::Tags::SqrtDetSpatialMetric< DataType >, gr::Tags::ExtrinsicCurvature< DataType, volume_dim >, gr::Tags::InverseSpatialMetric< DataType, volume_dim > > |
 Public Types inherited from gr::AnalyticSolution< Dim > | |
| template<typename DataType , typename Frame = ::Frame::Inertial> | |
| using | DerivLapse = ::Tags::deriv< gr::Tags::Lapse< DataType >, tmpl::size_t< volume_dim >, Frame > |
| template<typename DataType , typename Frame = ::Frame::Inertial> | |
| using | DerivShift = ::Tags::deriv< gr::Tags::Shift< DataType, volume_dim, Frame >, tmpl::size_t< volume_dim >, Frame > |
| template<typename DataType , typename Frame = ::Frame::Inertial> | |
| using | DerivSpatialMetric = ::Tags::deriv< gr::Tags::SpatialMetric< DataType, volume_dim, Frame >, tmpl::size_t< volume_dim >, Frame > |
| template<typename DataType , typename Frame = ::Frame::Inertial> | |
| using | tags = tmpl::list< gr::Tags::Lapse< DataType >, ::Tags::dt< gr::Tags::Lapse< DataType > >, DerivLapse< DataType, Frame >, gr::Tags::Shift< DataType, volume_dim, Frame >, ::Tags::dt< gr::Tags::Shift< DataType, volume_dim, Frame > >, DerivShift< DataType, Frame >, gr::Tags::SpatialMetric< DataType, volume_dim, Frame >, ::Tags::dt< gr::Tags::SpatialMetric< DataType, volume_dim, Frame > >, DerivSpatialMetric< DataType, Frame >, gr::Tags::SqrtDetSpatialMetric< DataType >, gr::Tags::ExtrinsicCurvature< DataType, volume_dim, Frame >, gr::Tags::InverseSpatialMetric< DataType, volume_dim, Frame > > |
Public Member Functions | |
| GaugeWave (double amplitude, double wavelength, const Options::Context &context={}) | |
| GaugeWave (const GaugeWave &)=default | |
| GaugeWave & | operator= (const GaugeWave &)=default |
| GaugeWave (GaugeWave &&)=default | |
| GaugeWave & | operator= (GaugeWave &&)=default |
| GaugeWave (CkMigrateMessage *) | |
| template<typename DataType , typename... Tags> | |
| tuples::TaggedTuple< Tags... > | variables (const tnsr::I< DataType, volume_dim, Frame::Inertial > &x, double t, tmpl::list< Tags... >) const |
| template<typename DataType , typename... Tags> | |
| tuples::TaggedTuple< Tags... > | variables (const tnsr::I< DataType, volume_dim, Frame::Inertial > &x, double t, const IntermediateVars< DataType > &vars, tmpl::list< Tags... >) const |
| void | pup (PUP::er &p) |
| double | amplitude () const |
| double | wavelength () const |
Static Public Attributes | |
| static constexpr size_t | volume_dim = Dim |
| static constexpr Options::String | help {"Gauge wave in flat spacetime"} |
| Static Public Attributes inherited from gr::AnalyticSolution< Dim > | |
| static constexpr size_t | volume_dim = Dim |
Detailed Description
class gr::Solutions::GaugeWave< Dim >
Gauge wave in flat spacetime.
Details
This solution is Minkowski space in coordinates chosen to contain a gauge wave. The spacetime metric is given by Eq. (4.3) of [2] :
\begin{equation} ds^2 = -H dt^2 + H dx^2 + dy^2 + dz^2, \end{equation}
where
\begin{equation} H = H(x-t) = 1 - A \sin\left(\frac{2\pi(x-t)}{d}\right). \end{equation}
The gauge wave has amplitude \(A\), wavelength \(d\), and propagates along the x axis.
In these coordinates, the spatial metric \(\gamma_{ij}\) and inverse spatial metric \(\gamma^{ij}\) are diagonal, with the diagonal elements equal to unity except for
\begin{align} \gamma_{xx} & = H,\\ \gamma^{xx} & = 1/H. \end{align}
The components of the derivatives of \(\gamma_{ij}\) vanish except for
\begin{align} \partial_t \gamma_{xx} & = \partial_t H = - \partial_x H,\\ \partial_x \gamma_{xx} & = \partial_x H. \end{align}
The square root of the spatial metric determinant is
\begin{align} \sqrt{\gamma} & = \sqrt{H}. \end{align}
The lapse and its derivatives are
\begin{align} \alpha & = \sqrt{H},\\ \partial_t \alpha & = -\frac{\partial_x H}{2\sqrt{H}},\\ \partial_x \alpha & = \frac{\partial_x H}{2\sqrt{H}},\\ \partial_y \alpha & = \partial_z \alpha = 0. \end{align}
The shift \(\beta^i\) and its derivatives vanish.
The extrinsic curvature's components vanish, except for
\begin{align} K_{xx} & = \frac{\partial_x H}{2 \sqrt{H}}. \end{align}
The following are input file options that can be specified:
The documentation for this class was generated from the following file:
- src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/GaugeWave.hpp