ForceFree::Solutions::ExactWald Class Reference

An exact electrovacuum force-free solution of Maxwell's equations in the Schwarzschild spacetime by Wald [204]. More...

#include <ExactWald.hpp>

Classes
struct	MagneticFieldAmplitude

Public Types
using	options = implementation defined

Public Member Functions
	ExactWald (const ExactWald &)=default
ExactWald &	operator= (const ExactWald &)=default
	ExactWald (ExactWald &&)=default
ExactWald &	operator= (ExactWald &&)=default
	ExactWald (double magnetic_field_amplitude)
auto	get_clone () const -> std::unique_ptr< evolution::initial_data::InitialData > override
void	pup (PUP::er &p) override
template<typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataVector, 3 > &x, const double t, tmpl::list< Tags... >) const
	Retrieve a collection of EM variables at (x, t)
template<typename Tag >
tuples::TaggedTuple< Tag >	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tag >) const
	Retrieve the metric variables.
virtual auto	get_clone () const -> std::unique_ptr< InitialData >=0

Static Public Attributes
static constexpr Options::String	help

Friends
bool	operator== (const ExactWald &lhs, const ExactWald &rhs)
auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildeE >) const -> tuples::TaggedTuple< Tags::TildeE >
	Retrieve the EM variables at (x,t).
auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildeB >) const -> tuples::TaggedTuple< Tags::TildeB >
	Retrieve the EM variables at (x,t).
static auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildePsi >) -> tuples::TaggedTuple< Tags::TildePsi >
	Retrieve the EM variables at (x,t).
static auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildePhi >) -> tuples::TaggedTuple< Tags::TildePhi >
	Retrieve the EM variables at (x,t).
static auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildeQ >) -> tuples::TaggedTuple< Tags::TildeQ >
	Retrieve the EM variables at (x,t).

Detailed Description

An exact electrovacuum force-free solution of Maxwell's equations in the Schwarzschild spacetime by Wald [204].

The solution is given in terms of the electromagnetic 4-potential

$$\begin{array}{}\text{(1)}& A_{\mu }=\frac{B_0}{2}({\varphi }_{\mu }+2a{t}_{\mu })\end{array}$$

where $B_0$ is the vector potential amplitude, ${\varphi }^{\mu }={\partial }_{\varphi }$, $t^{\mu }={\partial }_t$, and $a$ is the (dimensionless) spin of the black hole. The case $a=0$ is force-free outside the horizon.

Note

This solution is not force-free inside the horizon; the condition $E_i{B}^i=0$ is still satisfied, but $B^2>E^2$ is not.

In the spherical Kerr-Schild coordinates, the only nonzero component of vector potential is

$$\begin{array}{}\text{(2)}& A_{\varphi }=\frac{B_0}{2}{r}^2{\sin }^2\theta .\end{array}$$

Computing magnetic fields,

$$\begin{array}{}\text{(3)}& {\tilde{B}}^r& ={\partial }_{\theta }{A}_{\varphi }=B_0{r}^2\sin \theta \cos \theta \text{(4)}& {\tilde{B}}^{\theta }& =-{\partial }_r{A}_{\varphi }=-B_{0}r{\sin }^2\theta \text{(5)}& {\tilde{B}}^{\varphi }& =0,\end{array}$$

Transformation to the Cartesian coordinates gives

$$\begin{array}{}\text{(6)}& {\tilde{B}}^x=0,{\tilde{B}}^y=0,{\tilde{B}}^z=B_0.\end{array}$$

Electric fields are given by

$$\begin{array}{}\text{(7)}& E_i=F_{ia}{n}^a=\frac{1}{\alpha }(F_{i0}-F_{ij}{\beta }^j).\end{array}$$

We omit the derivation and write out results below:

$$\begin{array}{}\text{(8)}& {\tilde{E}}^x=-\frac{2M{B}_{0}y}{r^2},{\tilde{E}}^y=\frac{2M{B}_{0}x}{r^2},{\tilde{E}}^z=0\end{array}$$

Note that ${\tilde{B}}^i\equiv \sqrt{\gamma }{B}^i$, ${\tilde{E}}^i\equiv \sqrt{\gamma }{E}^i$, and $\gamma =1+2M/r$ in the (Cartesian) Kerr-Schild coordinates. We use $M=1$ Schwarzschild black hole in the Kerr-Schild coordinates (see the documentation of gr::Solutions::KerrSchild).

Member Function Documentation

get_clone()

auto ForceFree::Solutions::ExactWald::get_clone ( ) const -> std::unique_ptr< evolution::initial_data::InitialData >

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

help

constexpr Options::String ForceFree::Solutions::ExactWald::help

staticconstexpr

Initial value:

{
      "Exact vacuum solution of Maxwell's equations in Schwarzschild BH "
      "spacetime by Wald (1974)."}

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

• src/PointwiseFunctions/AnalyticSolutions/ForceFree/ExactWald.hpp