SpECTRE
v2024.05.11
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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). | |
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.
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).
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overridevirtual |
Implements evolution::initial_data::InitialData.
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staticconstexpr |