ForceFree::Solutions::ExactWald Class Reference

An exact electrovacuum force-free solution of Maxwell's equations in the Schwarzschild spacetime by Wald [218]. 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.
Public Member Functions inherited from evolution::initial_data::InitialData
virtual const InitialData &	unwrap () const
	Unwrap wrapper types (e.g. WithNoise, which wraps analytic initial data and adds random noise) to reach the underlying analytic solution or data.

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 [218].

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

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