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SpECTRE
v2026.04.01
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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 |
| template<typename DataType> | |
| using | DerivShift |
| template<typename DataType> | |
| using | DerivSpatialMetric |
| template<typename DataType> | |
| using | tags |
| Public Types inherited from gr::AnalyticSolution< Dim > | |
| template<typename DataType, typename Frame = ::Frame::Inertial> | |
| using | DerivLapse |
| template<typename DataType, typename Frame = ::Frame::Inertial> | |
| using | DerivShift |
| template<typename DataType, typename Frame = ::Frame::Inertial> | |
| using | DerivSpatialMetric |
| template<typename DataType, typename Frame = ::Frame::Inertial> | |
| using | tags |
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 |
Gauge wave in flat spacetime.
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:
| using gr::Solutions::GaugeWave< Dim >::DerivSpatialMetric |