ForceFree::Solutions::ExactWald Class Reference

An exact electrovacuum force-free solution of Maxwell's equations in the Schwarzschild spacetime by Wald [189]. More...

#include <ExactWald.hpp>

Classes
struct	MagneticFieldAmplitude

Public Types
using	options = tmpl::list< MagneticFieldAmplitude >

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 [189].

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

\begin{equation} A_\mu = \frac{B_0}{2}(\phi_\mu + 2a t_\mu) \end{equation}

where \(B_0\) is the vector potential amplitude, \(\phi^\mu = \partial_\phi\), \(t^\mu = \partial_t\), 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 \(E_iB^i = 0\) 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{equation} A_\phi = \frac{B_0}{2}r^2 \sin^2 \theta. \end{equation}

Computing magnetic fields,

\begin{align} \tilde{B}^r & = \partial_\theta A_\phi = B_0 r^2 \sin\theta\cos\theta \\ \tilde{B}^\theta & = - \partial_r A_\phi = -B_0 r \sin^2 \theta \\ \tilde{B}^\phi &= 0 , \end{align}

Transformation to the Cartesian coordinates gives

\begin{equation} \tilde{B}^x = 0 , \quad \tilde{B}^y = 0 , \quad \tilde{B}^z = B_0 . \end{equation}

Electric fields are given by

\begin{equation} E_i = F_{ia}n^a = \frac{1}{\alpha}(F_{i0} - F_{ij}\beta^j) . \end{equation}

We omit the derivation and write out results below:

\begin{equation} \tilde{E}^x = - \frac{2 M B_0 y}{r^2}, \quad \tilde{E}^y = \frac{2 M B_0 x}{r^2}, \quad \tilde{E}^z = 0 \end{equation}

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)."}

---

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

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