Kerr black hole in Spherical Kerr-Schild coordinates. More...

#include <SphericalKerrSchild.hpp>

Classes
struct	Center

class	IntermediateComputer

class	IntermediateVars

struct	internal_tags

struct	Mass

struct	Spin

Public Types
using	options = implementation defined

template<typename DataType , typename Frame = Frame::Inertial>
using	tags = implementation defined

template<typename DataType , typename Frame = ::Frame::Inertial>
using	CachedBuffer = CachedTempBuffer< internal_tags::x_minus_center< DataType, Frame >, internal_tags::r_squared< DataType >, internal_tags::r< DataType >, internal_tags::rho< DataType >, internal_tags::helper_matrix_F< DataType, Frame >, internal_tags::transformation_matrix_P< DataType, Frame >, internal_tags::jacobian< DataType, Frame >, internal_tags::helper_matrix_D< DataType, Frame >, internal_tags::helper_matrix_C< DataType, Frame >, internal_tags::deriv_jacobian< DataType, Frame >, internal_tags::transformation_matrix_Q< DataType, Frame >, internal_tags::helper_matrix_G1< DataType, Frame >, internal_tags::a_dot_x< DataType >, internal_tags::s_number< DataType >, internal_tags::helper_matrix_G2< DataType, Frame >, internal_tags::G1_dot_x< DataType, Frame >, internal_tags::G2_dot_x< DataType, Frame >, internal_tags::inv_jacobian< DataType, Frame >, internal_tags::helper_matrix_E1< DataType, Frame >, internal_tags::helper_matrix_E2< DataType, Frame >, internal_tags::deriv_inv_jacobian< DataType, Frame >, internal_tags::H< DataType >, internal_tags::kerr_schild_x< DataType, Frame >, internal_tags::a_cross_x< DataType, Frame >, internal_tags::kerr_schild_l< DataType, Frame >, internal_tags::l_lower< DataType, Frame >, internal_tags::l_upper< DataType, Frame >, internal_tags::deriv_r< DataType, Frame >, internal_tags::deriv_H< DataType, Frame >, internal_tags::kerr_schild_deriv_l< DataType, Frame >, internal_tags::deriv_l< DataType, Frame >, internal_tags::lapse_squared< DataType >, gr::Tags::Lapse< DataType >, internal_tags::deriv_lapse_multiplier< DataType >, internal_tags::shift_multiplier< DataType >, gr::Tags::Shift< DataType, 3, Frame >, DerivShift< DataType, Frame >, gr::Tags::SpatialMetric< DataType, 3, Frame >, DerivSpatialMetric< DataType, Frame >, ::Tags::dt< gr::Tags::SpatialMetric< DataType, 3, Frame > >, gr::Tags::ExtrinsicCurvature< DataType, 3, Frame >, gr::Tags::InverseSpatialMetric< DataType, 3, Frame >, gr::Tags::SpatialChristoffelFirstKind< DataType, 3, Frame >, gr::Tags::SpatialChristoffelSecondKind< DataType, 3, Frame > >

template<typename DataType , typename Frame = Frame::Inertial>
using	allowed_tags = implementation defined

Public Types inherited from gr::AnalyticSolution< 3_st >
using	DerivLapse = ::Tags::deriv< gr::Tags::Lapse< DataType >, tmpl::size_t< volume_dim >, Frame >

using	DerivShift = ::Tags::deriv< gr::Tags::Shift< DataType, volume_dim, Frame >, tmpl::size_t< volume_dim >, Frame >

using	DerivSpatialMetric = ::Tags::deriv< gr::Tags::SpatialMetric< DataType, volume_dim, Frame >, tmpl::size_t< volume_dim >, Frame >

using	tags = implementation defined

Public Member Functions
	SphericalKerrSchild (double mass, Spin::type dimensionless_spin, Center::type center, const Options::Context &context={})

	SphericalKerrSchild (CkMigrateMessage *)

	SphericalKerrSchild (const SphericalKerrSchild &)=default

SphericalKerrSchild &	operator= (const SphericalKerrSchild &)=default

	SphericalKerrSchild (SphericalKerrSchild &&)=default

SphericalKerrSchild &	operator= (SphericalKerrSchild &&)=default

void	pup (PUP::er &p)

double	mass () const

const std::array< double, volume_dim > &	center () const

const std::array< double, volume_dim > &	dimensionless_spin () const

template<typename DataType , typename Frame , typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, volume_dim, Frame > &x, double, tmpl::list< Tags... >) const

template<typename DataType , typename Frame , typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, volume_dim, Frame > &x, double, tmpl::list< Tags... >, gsl::not_null< IntermediateVars< DataType, Frame > * > cache) const

Static Public Attributes
static constexpr Options::String	help

Static Public Attributes inherited from gr::AnalyticSolution< 3_st >
static constexpr size_t	volume_dim

Detailed Description

Kerr black hole in Spherical Kerr-Schild coordinates.

Introduction

Given a Kerr-Schild (KS) black hole system, we denote the coordinate system using ${t, x, y, z}$ . In the transformed system, Spherical Kerr-Schild (Spherical KS), we denote the coordinate system using ${t, \bar{x}, \bar{y}, \bar{z}}$ . Further, when considering indexed objects, we will use Greek and Latin indices with the standard convention $(μ = 0, 1, 2, 3$ and $i = 1, 2, 3$ respectively), but we will use a bar to denote that a given index is in reference to the Spherical KS coordinate system (i.e. $i$ vs. $\bar{ı}$ and $μ$ vs. $\bar{μ}$ ).

Spin in the z direction

The Transformation

The Boyer-Lindquist radius for KS with its spin in the $z$ direction is defined by

$\begin{aligned} (1) & \frac{x^{2} + y^{2}}{r^{2} + a^{2}} + \frac{z^{2}}{r^{2}} = 1, \end{aligned}$

or equivalently,

$\begin{aligned} (2) & r^{2} & = \frac{1}{2} (x^{2} + y^{2} + z^{2} - a^{2}) + {(\frac{1}{4} {(x^{2} + y^{2} + z^{2} - a^{2})}^{2} + a^{2} z^{2})}^{1 / 2} . \end{aligned}$

The Spherical KS coordinates

$\begin{aligned} (3) & \vec{\bar{x}} = x^{\bar{ı}} = x_{\bar{ı}} = (\bar{x}, \bar{y}, \bar{z}), \end{aligned}$

and KS coordinates

$\begin{aligned} (4) & \vec{x} = x^{i} = x_{i} = (x, y, z), \end{aligned}$

are related by

$\begin{aligned} (5) & (\frac{\bar{x}}{r}, \frac{\bar{y}}{r}, \frac{\bar{z}}{r}) \equiv (\frac{x}{ρ}, \frac{y}{ρ}, \frac{z}{r}), \end{aligned}$

where we have defined

$\begin{aligned} (6) & ρ^{2} \equiv r^{2} + a^{2} . \end{aligned}$

Therefore, we have that $r$ satisfies the equation for a sphere in Spherical KS

$\begin{aligned} (7) & r^{2} = \vec{\bar{x}} \cdot \vec{\bar{x}} = {\bar{x}}^{2} + {\bar{y}}^{2} + {\bar{z}}^{2} . \end{aligned}$

It is clear to see that the Spherical KS radius coincides with the Boyer-Lindquist radius.

Spin in an Arbitrary Direction

Given that the remaining important quantities take forms that are easily specialized to the $z$ spin case, we instead focus on the general case for brevity.

The Transformation

The Boyer-Lindquist radius for KS with spin in an arbitrary direction is defined by

$\begin{aligned} (8) & r^{2} = \frac{1}{2} (\vec{x} \cdot \vec{x} - a^{2}) + {(\frac{1}{4} {(\vec{x} \cdot \vec{x} - a^{2})}^{2} + {(\vec{a} \cdot \vec{x})}^{2})}^{1 / 2} . \end{aligned}$

Then, defining two transformation matrices $Q^{\bar{ı}}_{j}$ and $P^{j}_{\bar{ı}}$ as

$\begin{aligned} (9) & Q^{\bar{ı}}_{j} & = \frac{r}{ρ} δ^{\bar{ı}}_{j} + \frac{1}{(ρ + r) ρ} a^{\bar{ı}} a_{j}, \\ (10) & P^{j}_{\bar{ı}} & = \frac{ρ}{r} δ^{j}_{\bar{ı}} - \frac{1}{(ρ + r) r} a^{j} a_{\bar{ı}}, \end{aligned}$

where the definition of $ρ$ is identical to Eq. $(6)$ , such that

$\begin{aligned} (11) & Q^{\bar{ı}}_{j} x^{j} & = x^{\bar{ı}}, \\ (12) & P^{j}_{\bar{ı}} x^{\bar{ı}} & = x^{j} . \end{aligned}$

We again recover that $r$ satisfies the equation for a sphere in Spherical KS

$\begin{aligned} (13) & r^{2} = \vec{\bar{x}} \cdot \vec{\bar{x}} = {\bar{x}}^{2} + {\bar{y}}^{2} + {\bar{z}}^{2}, \end{aligned}$

and we also have that

$\begin{aligned} (14) & \vec{a} \cdot \vec{\bar{x}} = \vec{a} \cdot \vec{x} . \end{aligned}$

Note that $Q^{\bar{ı}}_{j}$ and $P^{j}_{\bar{ı}}$ satisfy

$\begin{aligned} (15) & P^{i}_{\bar{ȷ}} Q^{\bar{ȷ}}_{k} = δ^{i}_{k} . \end{aligned}$

The Jacobian is then given by

$\begin{aligned} (16) & T^{i}_{\bar{ȷ}} = \partial_{\bar{ȷ}} x^{i} = \frac{\partial x^{i}}{\partial x^{\bar{ȷ}}}, \end{aligned}$

while its inverse is given by

$\begin{aligned} (17) & S^{\bar{ȷ}}_{i} = \partial_{i} x^{\bar{ȷ}} = \frac{\partial x^{\bar{ȷ}}}{\partial x^{i}}, \end{aligned}$

which in turn satisfies

$\begin{aligned} (18) & T^{i}_{\bar{ȷ}} S^{\bar{ȷ}}_{k} & = δ_{k}^{i} . \end{aligned}$

The Metric

A KS coordinate system is defined by

$\begin{aligned} (19) & g_{μ ν} = η_{μ ν} + 2 H l_{μ} l_{ν}, \end{aligned}$

where $H$ is a scalar function of the coordinates, $η_{μ ν}$ is the Minkowski metric, and $l^{μ}$ is a null vector. Note that the inverse of the spacetime metric is given by

$\begin{aligned} (20) & g^{μ ν} = η^{μ ν} - 2 H l^{μ} l^{ν} . \end{aligned}$

The scalar function $H$ takes the form

$\begin{aligned} (21) & H = \frac{M r^{3}}{r^{4} + {(\vec{a} \cdot \vec{x})}^{2}}, \end{aligned}$

where $M$ is the mass, while the spatial part of the null vector takes the form

$\begin{aligned} (22) & l_{i} = l^{i} = \frac{r \vec{x} - \vec{a} \times \vec{x} + \frac{(\vec{a} \cdot \vec{x}) \vec{a}}{r}}{ρ^{2}} . \end{aligned}$

Note that the full spacetime form of the null vector is

$\begin{aligned} (23) & l_{μ} & = (- 1, l_{i}), & l^{μ} & = (1, l^{i}) . \end{aligned}$

Transforming the KS spatial metric then yields the following Spherical KS spatial metric

$\begin{aligned} γ_{\bar{ı} \bar{ȷ}} & = γ_{m n} T^{m}_{\bar{ı}} T^{n}_{\bar{ȷ}}, \\ = η_{m n} T^{m}_{\bar{ı}} T^{n}_{\bar{ȷ}} + 2 H l_{m} l_{n} T^{m}_{\bar{ı}} T^{n}_{\bar{ȷ}}, \\ (24) & = η_{\bar{ı} \bar{ȷ}} + 2 H l_{\bar{ı}} l_{\bar{ȷ}} . \end{aligned}$

The transformed spacetime Minkowski metric is given by

$\begin{aligned} (25) & η_{\bar{μ} \bar{ν}} = (- 1) \otimes η_{\bar{ı} \bar{ȷ}}, \end{aligned}$

and the transformed spacetime null vector is given by

$\begin{aligned} (26) & l_{\bar{μ}} & = (- 1, l_{\bar{ı}}), & l^{\bar{μ}} & = (1, l^{\bar{ı}}) . \end{aligned}$

Therefore, the Spherical KS spacetime metric is

$\begin{aligned} (27) & g_{\bar{μ} \bar{ν}} = η_{\bar{μ} \bar{ν}} + 2 H l_{\bar{μ}} l_{\bar{ν}} . \end{aligned}$

Further, we have that the lapse in Spherical KS is given by

$\begin{aligned} (28) & α = {(1 + 2 H)}^{- 1 / 2}, \end{aligned}$

and the shift in Spherical KS by

$\begin{aligned} (29) & β^{\bar{ı}} & = - \frac{2 H l^{t} l^{\bar{ı}}}{1 + 2 H l^{t} l^{t}} = - 2 H α^{2} l^{t} l^{\bar{ı}}, & β_{\bar{ı}} & = - 2 H l_{t} l_{\bar{ı}} . \end{aligned}$

Derivatives

The derivatives of the preceding quantities are

$\begin{aligned} (30) & \frac{\partial r}{\partial x^{i}} & = \frac{r^{2} x_{i} + (\vec{a} \cdot \vec{x}) a_{i}}{r s}, \\ (31) & \partial_{\bar{ı}} H & = H T^{m}_{\bar{ı}} [\frac{3}{r} \frac{\partial r}{\partial x^{m}} - \frac{4 r^{3} \frac{\partial r}{\partial x^{m}} + 2 (\vec{a} \cdot \vec{x}) a_{m}}{r^{4} + {(\vec{a} \cdot \vec{x})}^{2}}], \\ \partial_{\bar{ȷ}} l^{\bar{ı}} & = \partial_{\bar{ȷ}} (l_{k} T^{k}_{\bar{ı}}), \\ (32) & = T^{k}_{\bar{ı}} T^{m}_{\bar{ȷ}} \frac{1}{ρ^{2}} [(x_{k} - 2 r l_{k} - \frac{(\vec{a} \cdot \vec{x}) a_{k}}{r^{2}}) \frac{\partial r}{\partial x^{m}} + r δ_{k m} + \frac{a_{k} a_{m}}{r} + ϵ^{k m n} a_{n}] + l_{k} \partial_{\bar{ȷ}} T^{k}_{\bar{ı}}, \\ (33) & \partial_{\bar{k}} γ_{\bar{ı} \bar{ȷ}} & = 2 l_{\bar{ı}} l_{\bar{ȷ}} \partial_{\bar{k}} H + 4 H l_{(\bar{ı}} \partial_{\bar{k}} l_{\bar{ȷ})} + T^{m}_{\bar{ȷ}} \partial_{\bar{k}} T^{m}_{\bar{ı}} + T^{m}_{\bar{ı}} \partial_{\bar{k}} T^{m}_{\bar{ȷ}}, \\ (34) & \partial_{\bar{k}} α & = - {(1 + 2 H)}^{- 3 / 2} \partial_{\bar{k}} H = - α^{3} \partial_{\bar{k}} H, \\ (35) & \partial_{\bar{k}} β^{i} & = 2 α^{2} [l^{\bar{ı}} \partial_{\bar{k}} H + H (S^{\bar{ı}}_{j} S^{\bar{m}}_{n} δ_{n j} \partial_{\bar{k}} l_{\bar{m}} + S^{\bar{ı}}_{j} l_{\bar{m}} \partial_{\bar{k}} S^{\bar{m}}_{n} δ_{n j} + S^{\bar{m}}_{n} δ_{n j} l_{\bar{m}} \partial_{\bar{k}} S^{\bar{ı}}_{j})] - 4 H l^{\bar{ı}} α^{4} \partial_{\bar{k}} H, \end{aligned}$

where we have defined $s$ as

$\begin{aligned} (36) & s & \equiv r^{2} + \frac{{(\vec{a} \cdot \vec{x})}^{2}}{r^{2}} . \end{aligned}$

Code

While the previous sections described the relevant physical quantities, the actual files make use of internally defined objects to ease the computation of the Jacobian, the inverse Jacobian, and their corresponding derivatives. Therefore, we now list these intermediary objects as well as how they construct the aforementioned physical quantities with the appropriate definition found in the code (all for arbitrary spin).

Helper Matrices

The intermediary objects used to define the various Jacobian objects, the so-called "helper matrices", are defined below.

$\begin{aligned} (37) & F^{i}_{\bar{k}} & \equiv - \frac{1}{ρ r^{3}} (a^{2} δ^{i}_{\bar{k}} - a^{i} a_{\bar{k}}), \\ (38) & {(G_{1})}^{\bar{ı}}_{\bar{m}} & \equiv \frac{1}{ρ^{2} r} (a^{2} δ^{\bar{ı}}_{\bar{m}} - a^{\bar{ı}} a_{\bar{m}}), \\ (39) & {(G_{2})}^{\bar{n}}_{j} & \equiv \frac{ρ^{2}}{s r} Q^{\bar{n}}_{j}, \\ (40) & D^{i}_{\bar{m}} & \equiv \frac{1}{ρ^{3} r} (a^{2} δ^{i}_{\bar{m}} - a^{i} a_{\bar{m}}), \\ (41) & C^{i}_{\bar{m}} & \equiv D^{i}_{\bar{m}} - 3 F^{i}_{\bar{m}} = \frac{1}{ρ r} (\frac{1}{ρ^{2}} + \frac{3}{r^{2}}) (a^{2} δ^{i}_{\bar{m}} - a^{i} a_{\bar{m}}), \\ {(E_{1})}^{i}_{\bar{m}} & \equiv - \frac{1}{ρ^{2}} (\frac{1}{r^{2}} + \frac{2}{ρ^{2}}) (a^{2} δ^{i}_{\bar{m}} - a^{i} a_{\bar{m}}), \\ (42) & = - \frac{(ρ^{2} + 2 r^{2})}{r^{2} ρ^{4}} (a^{2} δ^{i}_{\bar{m}} - a^{i} a_{\bar{m}}), \\ (43) & {(E_{2})}^{\bar{n}}_{j} & \equiv [- \frac{a^{2}}{ρ^{2} r} - \frac{2}{s} (r - \frac{{(\vec{a} \cdot \vec{x})}^{2}}{r^{3}})] \cdot {(G_{2})}^{\bar{n}}_{j} + \frac{1}{s} P^{\bar{n}}_{j}, \end{aligned}$

where $s$ is defined identically to Eq. $(36)$ .

Physical Quantities

Below are the definitions for how we construct the Jacobian, inverse Jacobian, derivative of the Jacobian, and derivative of the inverse Jacobian in the code using the helper matrices.

$\begin{aligned} (44) & T^{i}_{\bar{ȷ}} & = P^{i}_{\bar{ȷ}} + F^{i}_{\bar{k}} x^{\bar{k}} x_{\bar{ȷ}}, \\ (45) & S^{\bar{ı}}_{j} & = Q^{\bar{ı}}_{j} + {(G_{1})}^{\bar{ı}}_{\bar{m}} x^{\bar{m}} x_{\bar{n}} {(G_{2})}^{\bar{n}}_{j}, \\ (46) & \partial_{\bar{k}} T^{i}_{\bar{ȷ}} & = F^{i}_{\bar{ȷ}} x_{\bar{k}} + F^{i}_{\bar{k}} x_{\bar{ȷ}} + F^{i}_{\bar{m}} x^{\bar{m}} δ_{j k} + C^{i}_{\bar{m}} \frac{x_{\bar{k}} x^{\bar{m}} x_{\bar{ȷ}}}{r^{2}}, \\ \partial_{\bar{k}} S^{\bar{ı}}_{j} & = D^{\bar{ı}}_{j} x_{\bar{k}} + {(G_{1})}^{\bar{ı}}_{\bar{k}} x_{\bar{n}} {(G_{2})}^{\bar{n}}_{j} + {(G_{1})}^{\bar{ı}}_{\bar{m}} x^{\bar{m}} {(G_{2})}^{\bar{n}}_{j} δ_{\bar{n} \bar{k}} \\ (47) & + {(E_{1})}^{i}_{\bar{m}} \frac{x_{\bar{k}} x^{\bar{m}} x_{\bar{n}}}{r} {(G_{2})}^{\bar{n}}_{j} + {(G_{1})}^{\bar{i}}_{\bar{m}} \frac{x_{\bar{k}} x^{\bar{m}} x_{\bar{n}}}{r} {(E_{2})}^{\bar{n}}_{j} - {(G_{1})}^{\bar{ı}}_{\bar{m}} x^{\bar{m}} x_{\bar{n}} {(G_{2})}^{\bar{n}}_{j} \frac{2 (\vec{a} \cdot \vec{x})}{s r^{2}} a_{\bar{k}} . \end{aligned}$

Member Data Documentation

◆ help

constexpr Options::String gr::Solutions::SphericalKerrSchild::help

staticconstexpr

Initial value:

{

"Black hole in Spherical Kerr-Schild coordinates"}

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

src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/SphericalKerrSchild.hpp

Classes

Public Types

Public Member Functions

Static Public Attributes

Detailed Description

Introduction

Spin in the z direction

The Transformation

Spin in an Arbitrary Direction

The Transformation

The Metric

Derivatives

Code

Helper Matrices

Physical Quantities

Member Data Documentation

◆ help