gr::Solutions::GaugePlaneWave< Dim > Class Template Reference

Gauge plane wave in flat spacetime. More...

#include <GaugePlaneWave.hpp>

Classes
struct	IntermediateVars
struct	Profile
struct	WaveVector

Public Types
using	options = implementation defined
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 >
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 = implementation defined

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)

Detailed Description

template<size_t Dim>
class gr::Solutions::GaugePlaneWave< Dim >

Gauge plane wave in flat spacetime.

Details

The spacetime metric is in Kerr-Schild form:

$$\begin{array}{}\text{(1)}& g_{\mu \nu }={\eta }_{\mu \nu }+H{l}_{\mu }{l}_{\nu }\end{array}$$

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

$$\begin{array}{}\text{(2)}& g^{\mu \nu }\equiv {\eta }^{\mu \nu }-H{l}^{\mu }{l}^{\nu },\end{array}$$

and that $\sqrt{-g}=1$. Also, $l_{\mu }$ is a geodesic with respect to both the physical metric and the Minkowski metric:

$$\begin{array}{}\text{(3)}& l^{\mu }{\partial }_{\mu }{l}_{\nu }=l^{\mu }{\mathrm{\nabla }}_{\mu }{l}_{\nu }=0.\end{array}$$

For this solution we choose the profile $H$ of the plane wave to be an arbitrary one-dimensional function of $u=\overrightarrow{k}\cdot \overrightarrow{x}-\omega t$ with a constant Euclidean wave vector $\overrightarrow{k}$ and frequency $\omega =||\overrightarrow{k}||$. The null covector is chosen to be $l_a=(-\omega ,k_i)$. Thus, if $H=H[u]$, then ${\partial }_{\mu }H=H^{\prime }[u]l_{\mu }$. Therefore the derivatives of the spacetime metric are:

$$\begin{array}{}\text{(4)}& {\partial }_{\rho }{g}_{\mu \nu }=H^{\prime }{l}_{\rho }{l}_{\mu }{l}_{\nu },\end{array}$$

The 3+1 quantities are

$$\begin{array}{}\text{(5)}& \alpha & ={(1+H{\omega }^2)}^{-1/2},\text{(6)}& {\beta }^i& =\frac{-H\omega k^i}{1+H{\omega }^2},\text{(7)}& {\gamma }_{ij}& ={\delta }_{ij}+H{k}_i{k}_j,\text{(8)}& \gamma & =1+H{\omega }^2,\text{(9)}& {\gamma }^{ij}& ={\delta }^{ij}-\frac{H{k}^i{k}^j}{1+H{\omega }^2},\text{(10)}& {\partial }_t\alpha & =\frac{{\omega }^3{H}^{\prime }}{2{(1+H{\omega }^2)}^{3/2}},\text{(11)}& {\partial }_i\alpha & =-\frac{{\omega }^2{H}^{\prime }{k}_i}{2{(1+H{\omega }^2)}^{3/2}},\text{(12)}& {\partial }_t{\beta }^i& =\frac{{\omega }^2{H}^{\prime }{k}^i}{{(1+H{\omega }^2)}^2},\text{(13)}& {\partial }_j{\beta }^i& =-\frac{\omega H^{\prime }{k}_j{k}^i}{{(1+H{\omega }^2)}^2},\text{(14)}& {\partial }_t{\gamma }_{ij}& =-\omega H^{\prime }{k}_i{k}_j,\text{(15)}& {\partial }_k{\gamma }_{ij}& =H^{\prime }{k}_k{k}_i{k}_j,\text{(16)}& K_{ij}& =-\frac{\omega H^{\prime }{k}_i{k}_j}{2{(1+H{\omega }^2)}^{1/2}}.\end{array}$$

Note that this solution is a gauge wave as ${\mathrm{\Gamma }}^a{}_{bc}=\frac{1}{2}{H}^{\prime }{l}^a{l}_b{l}_c$ and thus $R^a{}_{bcd}=0$.

Template Parameters

Dim	the spatial dimension of the solution

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

• src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/GaugePlaneWave.hpp