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SpECTRE
v2026.04.01
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Gauge plane wave in flat spacetime. More...
#include <GaugePlaneWave.hpp>
Classes | |
| struct | IntermediateVars |
| struct | Profile |
| struct | WaveVector |
Public Types | |
| using | options = tmpl::list<WaveVector, Profile> |
| template<typename DataType> | |
| using | DerivLapse |
| template<typename DataType> | |
| using | DerivShift |
| template<typename DataType> | |
| using | DerivSpatialMetric |
| 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 | |
| GaugePlaneWave (const std::array< double, Dim > &wave_vector, std::unique_ptr< MathFunction< 1, Frame::Inertial > > profile) | |
| GaugePlaneWave (const GaugePlaneWave &) | |
| GaugePlaneWave & | operator= (const GaugePlaneWave &) |
| GaugePlaneWave (GaugePlaneWave &&)=default | |
| GaugePlaneWave & | operator= (GaugePlaneWave &&)=default |
| GaugePlaneWave (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 |
| std::array< double, Dim > | get_wave_vector () const |
| void | pup (PUP::er &p) |
Static Public Attributes | |
| static constexpr size_t | volume_dim = Dim |
| static constexpr Options::String | help {"Gauge plane wave in flat spacetime"} |
| Static Public Attributes inherited from gr::AnalyticSolution< Dim > | |
| static constexpr size_t | volume_dim = Dim |
Friends | |
| template<size_t LocalDim> | |
| bool | operator== (const GaugePlaneWave< LocalDim > &lhs, const GaugePlaneWave< LocalDim > &rhs) |
| template<size_t LocalDim> | |
| bool | operator!= (const GaugePlaneWave< LocalDim > &lhs, const GaugePlaneWave< LocalDim > &rhs) |
Gauge plane wave in flat spacetime.
The spacetime metric is in Kerr-Schild form:
\begin{equation}g_{\mu\nu} = \eta_{\mu\nu} + H l_\mu l_\nu \end{equation}
where \(H\) is a scalar function of the coordinates, \(\eta_{\mu\nu}\) is the Minkowski metric, and \(l^\mu\) is a null vector. Note that the form of the metric along with the nullness of \(l^\mu\) allows one to raise and lower indices of \(l^\mu\) using \(\eta_{\mu\nu}\), and that \(l^t l^t = l_t l_t = l^i l_i\). Note also that
\begin{equation}g^{\mu\nu} \equiv \eta^{\mu\nu} - H l^\mu l^\nu, \end{equation}
and that \(\sqrt{-g}=1\). Also, \(l_\mu\) is a geodesic with respect to both the physical metric and the Minkowski metric:
\begin{equation}l^\mu \partial_\mu l_\nu = l^\mu\nabla_\mu l_\nu = 0. \end{equation}
For this solution we choose the profile \(H\) of the plane wave to be an arbitrary one-dimensional function of \(u = \vec{k} \cdot \vec{x} - \omega t\) with a constant Euclidean wave vector \(\vec{k}\) and frequency \(\omega = ||\vec{k}||\). The null covector is chosen to be \(l_a = (-\omega, k_i)\). Thus, if \(H = H[u]\), then \(\partial_\mu H = H'[u] l_\mu\). Therefore the derivatives of the spacetime metric are:
\begin{equation}\partial_\rho g_{\mu\nu} = H' l_\rho l_\mu l_\nu, \end{equation}
The 3+1 quantities are
\begin{align}\alpha & = \left( 1 + H \omega^2 \right)^{-1/2},\\ \beta^i & = \frac{-H \omega k^i}{1 + H \omega^2},\\ \gamma_{ij} & = \delta_{ij} + H k_i k_j,\\ \gamma & = 1 + H \omega^2,\\ \gamma^{ij} & = \delta^{ij} - \frac{H k^i k^j}{1 + H \omega^2},\\ \partial_t \alpha & = \frac{\omega^3 H'}{2 \left(1 + H \omega^2 \right)^{3/2}},\\ \partial_i \alpha & = - \frac{\omega^2 H' k_i}{2 \left(1 + H \omega^2 \right)^{3/2}},\\ \partial_t \beta^i & = \frac{\omega^2 H' k^i}{\left(1 + H \omega^2 \right)^2},\\ \partial_j \beta^i & = - \frac{\omega H' k_j k^i}{\left(1 + H \omega^2 \right)^2},\\ \partial_t \gamma_{ij} & = - \omega H' k_i k_j,\\ \partial_k \gamma_{ij} & = H' k_k k_i k_j,\\ K_{ij} & = - \frac{\omega H' k_i k_j}{2 \left(1 + H \omega^2 \right)^{1/2}}. \end{align}
Note that this solution is a gauge wave as \(\Gamma^a{}_{bc} = \frac{1}{2} H' l^a l_b l_c\) and thus \(R^a{}_{bcd} = 0\).
| Dim | the spatial dimension of the solution |
| using gr::Solutions::GaugePlaneWave< Dim >::DerivLapse |
| using gr::Solutions::GaugePlaneWave< Dim >::DerivShift |
| using gr::Solutions::GaugePlaneWave< Dim >::DerivSpatialMetric |