domain::CoordinateMaps::KerrHorizonConforming Class Reference

Distorts cartesian coordinates $x^i$ such that a coordinate sphere ${\delta }_{ij}{x}^i{x}^j=C^2$ is mapped to an ellipsoid of constant Kerr-Schild radius $r=C$. More...

#include <KerrHorizonConforming.hpp>

Public Member Functions
	KerrHorizonConforming (const double mass, std::array< double, 3 > dimensionless_spin)
	Constructs a Kerr horizon conforming map. More...
template<typename T >
std::array< tt::remove_cvref_wrap_t< T >, 3 >	operator() (const std::array< T, 3 > &source_coords) const
std::optional< std::array< double, 3 > >	inverse (const std::array< double, 3 > &target_coords) const
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 3, Frame::NoFrame >	jacobian (const std::array< T, 3 > &source_coords) const
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 3, Frame::NoFrame >	inv_jacobian (const std::array< T, 3 > &source_coords) const
bool	is_identity () const
void	pup (PUP::er &p)

Static Public Attributes
static constexpr size_t	dim = 3

Friends
bool	operator== (const KerrHorizonConforming &lhs, const KerrHorizonConforming &rhs)

Detailed Description

Distorts cartesian coordinates $x^i$ such that a coordinate sphere ${\delta }_{ij}{x}^i{x}^j=C^2$ is mapped to an ellipsoid of constant Kerr-Schild radius $r=C$.

The Kerr-Schild radius $r$ is defined as the largest positive root of

$$r^4-r^2(x^2-a^2)-(\overrightarrow{a}\cdot \overrightarrow{x}{)}^2=0.$$

In this equation, $\overrightarrow{x}$ are the coordinates of the (distorted) surface, and $\overrightarrow{a}=\overrightarrow{S}/M$ is the spin-parameter of the black hole. This is equivalent to the implicit definition of $r$ in Eq. (47) of [130]. The black hole is assumed to be at the origin.

Details

Given a spin vector $\overrightarrow{a}$, we define the Kerr-Schild radius:

$$\begin{array}{}\text{(1)}& r(\overrightarrow{x})=\sqrt{\frac{x^2-a^2+\sqrt{(x^2-a^2{)}^2+4(\overrightarrow{x}\cdot \overrightarrow{a}{)}^2}}{2}}\end{array}$$

We also define the auxiliary variable:

$$\begin{array}{}\text{(2)}& s(\overrightarrow{x})=\frac{x^2(x^2+a^2)}{x^4+(\overrightarrow{x}\cdot \overrightarrow{a}{)}^2}\end{array}$$

The map is then given by:

$$\begin{array}{}\text{(3)}& \overrightarrow{x}(\overrightarrow{\xi })=\overrightarrow{\xi }\sqrt{s(\overrightarrow{\xi })}\end{array}$$

The inverse map is given by:

$$\begin{array}{}\text{(4)}& \overrightarrow{\xi }(\overrightarrow{x})=\overrightarrow{x}\frac{r(\overrightarrow{x})}{|\overrightarrow{x}|}\end{array}$$

The jacobian is:

$$\begin{array}{}\text{(5)}& \frac{\partial {\xi }^i}{\partial x^j}={\delta }_{ij}\sqrt{s(\overrightarrow{\xi })}+\frac{{\xi }_j}{2\sqrt{s(\overrightarrow{\xi })}}\frac{4{\xi }^2(1-s(\overrightarrow{\xi })){\xi }_j+2a^2{\xi }_i+2s(\overrightarrow{\xi })(\overrightarrow{\xi }\cdot \overrightarrow{a})a_i}{{\xi }^4+(\overrightarrow{\xi }\cdot \overrightarrow{a}{)}^2}\end{array}$$

The inverse jacobian is:

$$\begin{array}{}\text{(6)}& \frac{\partial x^i}{\partial {\xi }^j}={\delta }_{ij}\frac{r(\overrightarrow{x})}{|\overrightarrow{x}|}+\frac{r(\overrightarrow{x}{)}^2{x}_i+(\overrightarrow{x}\cdot \overrightarrow{a})a_i}{2r(\overrightarrow{x}{)}^3-r(\overrightarrow{x})(x^2-a^2)}\frac{x_j}{|\overrightarrow{x}|}-\frac{r(\overrightarrow{x})x_i{x}_j}{x^3}\end{array}$$

Constructor & Destructor Documentation

KerrHorizonConforming()

domain::CoordinateMaps::KerrHorizonConforming::KerrHorizonConforming ( const double  mass, std::array< double, 3 >  dimensionless_spin  )

explicit

Constructs a Kerr horizon conforming map.

Parameters

mass	The Kerr mass parameter $M$
dimensionless_spin	The dimensionless spin $\overrightarrow{\chi }=\overrightarrow{a}/M=\overrightarrow{S}/M^2$, where $M$ is the Kerr mass parameter, $\overrightarrow{S}$ is the angular momentum or quasilocal spin, and $\overrightarrow{a}$ is the Kerr spin parameter.

Note

The horizon depends only on the dimensionful spin parameter $\overrightarrow{a}=M\overrightarrow{\chi }$. This constructor takes $M$ and $\overrightarrow{\chi }$ separately for consistency with other code such as gr::Solutions::KerrSchild, and hence to avoid bugs where the wrong spin quantity is used accidentally.

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

• src/Domain/CoordinateMaps/KerrHorizonConforming.hpp