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

Gauge wave in flat spacetime. More...

#include <GaugeWave.hpp>

Classes
struct	Amplitude
struct	Wavelength

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 >
template<typename DataType >
using	tags = implementation defined
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
	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

template<size_t Dim>
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] :

$$\begin{array}{}\text{(1)}& d{s}^2=-Hd{t}^2+Hd{x}^2+d{y}^2+d{z}^2,\end{array}$$

where

$$\begin{array}{}\text{(2)}& H=H(x-t)=1-A\sin \left(\frac{2\pi (x-t)}{d}\right).\end{array}$$

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

$$\begin{array}{}\text{(3)}& {\gamma }_{xx}& =H,\text{(4)}& {\gamma }^{xx}& =1/H.\end{array}$$

The components of the derivatives of ${\gamma }_{ij}$ vanish except for

$$\begin{array}{}\text{(5)}& {\partial }_t{\gamma }_{xx}& ={\partial }_{t}H=-{\partial }_{x}H,\text{(6)}& {\partial }_x{\gamma }_{xx}& ={\partial }_{x}H.\end{array}$$

The square root of the spatial metric determinant is

$$\begin{array}{}\text{(7)}& \sqrt{\gamma }& =\sqrt{H}.\end{array}$$

The lapse and its derivatives are

$$\begin{array}{}\text{(8)}& \alpha & =\sqrt{H},\text{(9)}& {\partial }_t\alpha & =-\frac{{\partial }_{x}H}{2\sqrt{H}},\text{(10)}& {\partial }_x\alpha & =\frac{{\partial }_{x}H}{2\sqrt{H}},\text{(11)}& {\partial }_y\alpha & ={\partial }_z\alpha =0.\end{array}$$

The shift ${\beta }^i$ and its derivatives vanish.

The extrinsic curvature's components vanish, except for

$$\begin{array}{}\text{(12)}& K_{xx}& =\frac{{\partial }_{x}H}{2\sqrt{H}}.\end{array}$$

The following are input file options that can be specified:

• Amplitude
• Wavelength

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

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