Binary black holes system. More...

#include <bhole.h>

Public Member Functions
	Bhole_binaire (Map_af &mp1, Map_af &mp2)
	Standard constructor. More...

	Bhole_binaire (const Bhole_binaire &)
	Copy. More...

	~Bhole_binaire ()
	Destructor. More...

void	operator= (const Bhole_binaire &)
	Affectation operator. More...

Bhole &	set (int i)
	Read/write of a component of the system. More...

void	set_omega (double ome)
	Sets the orbital velocity to `ome`. More...

void	set_pos_axe (double)

const Bhole &	operator() (int i) const
	Read only of a component of the system. More...

double	get_omega () const
	Returns the angular velocity. More...

void	init_bhole_binaire ()
	Initialisation of the system. More...

double	viriel () const
	Computes the viriel error, that is the difference between the ADM and the Komar masses, calculated by the asymptotic behaviours of respectively $\Psi$ and N . More...

void	solve_lapse (double precis, double relax)
	Solves the equation for the lapse~: $\Delta N_a = -\frac{2}{\Psi}\nabla_i \Psi_a \nabla^i N + N \Psi^4 K_{ij}K_a^{ij}$ The fields are the total values excpet those with subscript , which are the fields generated by each holes (a = 1, 2). More...

void	solve_psi (double precis, double relax)
	Solves the equation for the conformal factor~: $*\Delta \Psi_a = -\frac {\Psi^5}{8} K_{ij}K_a^{ij}$ The fields are the total values excpet those with subscript , which are the fields generated by each holes (a = 1, 2). More...

void	solve_shift (double precis, double relax)
	Solves the equation for the shift, using the Oohara-Nakarmure scheme~: $*\Delta \beta_a^i +\frac{1}{3} \nabla^i \left(\nabla_j \beta_a^j\right) = -6A^{ij}\frac{\nabla_i \Psi_a}{\Psi} + 2K^{ij}\nabla_j N_a$ using the current $\Omega$ for the boudary conditions~: $\vec{N} = -\Omega \vec{m}$ on both horizons. More...

void	fait_tkij ()
	Calculation af the extrinsic curvature tensor. More...

void	fait_decouple ()
	Calculates {tt decouple} which is used to obtain `tkij_auto` by the formula : `tkij_auto` = `decouple` * `tkij_tot` . More...

void	set_statiques (double precis, double relax)
	Initialize the systeme to Misner Lindquist solution, that is solving for N and in the case $\Omega = 0$ . More...

void	coal (double precis, double relax, int nbre_ome, double search_ome, double m1, double m2, const int sortie=0)
	Solves the equation for a particular angular velocity, the systeme being initialized to Misner-Lindquist solution. More...

double	adm_systeme () const
	Calculates the ADM mass of the system using : $M = -\frac{1}{2 \pi} \oint_{\infty} \nabla^i \Psi {\mathrm d} S_i$ . More...

double	komar_systeme () const
	Calculates the Komar mass of the system using : $M = \frac{1}{4 \pi} \oint_{\infty} \nabla^i N {\mathrm d} S_i$ . More...

double	moment_systeme_inf ()
	Calculates the angular momentum of the black hole using the formula at infinity : $J = -\frac{1}{8 \pi} \oint_{\infty} K_j^i m^j {\mathrm d}S_i$ where $\vec{m} = \frac{\partial}{\partial \phi}$ . More...

double	moment_systeme_hor () const
	Calculates the angular momentum of the black hole using the formula on the horizon : $J = -\sum_{a= 1, 2} \frac{1}{8 \pi} \oint_{H_a} \Psi^6 K_j^i m^j {\mathrm d}S_i$ where $\vec{m} = \frac{\partial}{\partial \phi}$ and denotes the horizon of the hole a . More...

double	distance_propre (const int nr=65) const
	Calculation of the proper distance between the two spheres of inversion, along the x axis. More...

Tbl	linear_momentum_systeme_inf () const

void	solve_phi (double precision, double relax)
	Solve the equation for the logarithm of $\Psi$ . More...

void	init_phi ()
	Initiates the system for a resolution using the logarithm of $\Psi$ . More...

Private Attributes
Bhole	hole1
	Black hole one. More...

Bhole	hole2
	Black hole two. More...

Bhole *	holes [2]
	Array on the black holes. More...

double	pos_axe

double	omega
	Position of the axis of rotation. More...

Detailed Description

Binary black holes system.

()

This class is intended for dealing with binary black holes configurations in the conformaly flat approximation.

Definition at line 757 of file bhole.h.

Constructor & Destructor Documentation

◆ Bhole_binaire() [1/2]

Lorene::Bhole_binaire::Bhole_binaire	(	Map_af &	mp1,
		Map_af &	mp2
	)

Standard constructor.

Definition at line 214 of file bhole_binaire.C.

References hole1, hole2, and holes.

◆ Bhole_binaire() [2/2]

Lorene::Bhole_binaire::Bhole_binaire ( const Bhole_binaire & source )

Copy.

Definition at line 222 of file bhole_binaire.C.

References hole1, hole2, and holes.

◆ ~Bhole_binaire()

Lorene::Bhole_binaire::~Bhole_binaire ( )

Destructor.

Definition at line 229 of file bhole_binaire.C.

Member Function Documentation

◆ adm_systeme()

double Lorene::Bhole_binaire::adm_systeme ( ) const

Calculates the ADM mass of the system using : $M = -\frac{1}{2 \pi} \oint_{\infty} \nabla^i \Psi {\mathrm d} S_i$ .

Definition at line 89 of file bhole_glob.C.

References hole1, hole2, Lorene::Map_af::integrale_surface_infini(), Lorene::Bhole::mp, and Lorene::Bhole::psi_auto.

◆ coal()

void Lorene::Bhole_binaire::coal	(	double	precis,
		double	relax,
		int	nbre_ome,
		double	search_ome,
		double	m1,
		double	m2,
		const int	sortie = `0`
	)

Solves the equation for a particular angular velocity, the systeme being initialized to Misner-Lindquist solution.

Parameters

precis	[input] : precision for the convergence (on $\beta$ ).
relax	[input] : relaxation parameter.
nbre_ome	[input] : number of intermediates velocities to go from 0 to `omega` , typically 10.
sortie	[input] : flag for the output on files (0 no output files).

Returns: : the virial error.

Definition at line 160 of file bhole_coal.C.

References Lorene::Bhole::area(), Lorene::diffrelmax(), fait_tkij(), Lorene::Map::get_mg(), Lorene::Mg3d::get_nzone(), Lorene::Map::get_ori_x(), Lorene::Bhole::get_regul(), Lorene::Bhole::get_shift_auto(), hole1, hole2, Lorene::Map_af::homothetie_interne(), Lorene::Bhole::mp, omega, Lorene::pow(), Lorene::Bhole::rayon, Lorene::Bhole::regul, set_omega(), solve_lapse(), solve_psi(), solve_shift(), Lorene::sqrt(), Lorene::Map_af::val_r(), and viriel().

◆ distance_propre()

double Lorene::Bhole_binaire::distance_propre ( const int nr = 65 ) const

Calculation of the proper distance between the two spheres of inversion, along the x axis.

Parameters

nr	[input] : number of points used for the calculation.

Definition at line 265 of file bhole_glob.C.

References Lorene::Map::convert_absolute(), Lorene::cos(), Lorene::Map::get_ori_x(), hole1, hole2, Lorene::Bhole::mp, Lorene::pow(), Lorene::Bhole::psi_auto, and Lorene::Bhole::rayon.

◆ fait_decouple()

void Lorene::Bhole_binaire::fait_decouple ( )

Calculates {tt decouple} which is used to obtain tkij_auto by the formula : tkij_auto = decouple * tkij_tot .

(see the membre {tt Cmp decouple for more precisions about its value).

Definition at line 317 of file bhole_kij.C.

References Lorene::Cmp::allocate_all(), Lorene::Map::convert_absolute(), Lorene::cos(), Lorene::Bhole::decouple, Lorene::Map::get_mg(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Map::get_ori_x(), hole1, hole2, Lorene::Bhole::mp, Lorene::pow(), Lorene::Map::r, Lorene::Cmp::set(), Lorene::sin(), Lorene::Cmp::std_base_scal(), Lorene::Cmp::val_point(), Lorene::Map::xa, Lorene::Map::ya, and Lorene::Map::za.

◆ fait_tkij()

void Lorene::Bhole_binaire::fait_tkij ( )

Calculation af the extrinsic curvature tensor.

The regularisation of the shifts must have been done. All the contributions of $A^{ij}$ are then calculated and the total tensor must be zero on both horizons. The computation is done to avoid every singularity close to the horizons (division by N =0) for it is done in the coefficient space in the two regions surrounding the holes.

Definition at line 91 of file bhole_kij.C.

References Lorene::Tenseur::dec2_dzpuis(), Lorene::Bhole::fait_taij_auto(), Lorene::Tenseur::get_etat(), Lorene::Tenseur::get_mp(), Lorene::Map::get_rot_phi(), hole1, hole2, Lorene::Tenseur::set_etat_qcq(), Lorene::Bhole::taij_auto, Lorene::Bhole::taij_comp, and Lorene::Bhole::tkij_tot.

◆ get_omega()

double Lorene::Bhole_binaire::get_omega ( ) const

inline

Returns the angular velocity.

Definition at line 808 of file bhole.h.

References omega.

◆ init_bhole_binaire()

void Lorene::Bhole_binaire::init_bhole_binaire ( )

Initialisation of the system.

Each hole is set close to a Schwarzschild one and the parts of the fields generated by the companion are calculated.

The angular velocity is set to zero.

Definition at line 242 of file bhole_binaire.C.

References fait_decouple(), Lorene::Bhole::fait_n_comp(), Lorene::Bhole::fait_psi_comp(), hole1, hole2, Lorene::Bhole::init_bhole(), and set_omega().

◆ init_phi()

void Lorene::Bhole_binaire::init_phi ( )

Initiates the system for a resolution using the logarithm of $\Psi$ .

Definition at line 158 of file bhole_solve_phi.C.

References Lorene::Bhole::fait_psi_comp(), hole1, hole2, Lorene::Bhole::init_bhole_phi(), and set_omega().

◆ komar_systeme()

double Lorene::Bhole_binaire::komar_systeme ( ) const

Calculates the Komar mass of the system using : $M = \frac{1}{4 \pi} \oint_{\infty} \nabla^i N {\mathrm d} S_i$ .

Definition at line 100 of file bhole_glob.C.

References hole1, hole2, Lorene::Map_af::integrale_surface_infini(), Lorene::Bhole::mp, and Lorene::Bhole::n_auto.

◆ moment_systeme_hor()

double Lorene::Bhole_binaire::moment_systeme_hor ( ) const

Calculates the angular momentum of the black hole using the formula on the horizon : $J = -\sum_{a= 1, 2} \frac{1}{8 \pi} \oint_{H_a} \Psi^6 K_j^i m^j {\mathrm d}S_i$ where $\vec{m} = \frac{\partial}{\partial \phi}$ and $H_a$ denotes the horizon of the hole a .

Definition at line 111 of file bhole_glob.C.

References Lorene::Map::get_bvect_cart(), Lorene::Map::get_bvect_spher(), Lorene::Map::get_mg(), Lorene::Mg3d::get_nzone(), Lorene::Map::get_rot_phi(), hole1, hole2, Lorene::Map_af::integrale_surface(), Lorene::Bhole::mp, omega, Lorene::pow(), Lorene::Bhole::psi_tot, Lorene::Bhole::rayon, Lorene::Tenseur::set_etat_qcq(), Lorene::Tenseur::set_std_base(), Lorene::Cmp::std_base_scal(), Lorene::Bhole::tkij_tot, Lorene::Map::xa, and Lorene::Map::ya.

◆ moment_systeme_inf()

double Lorene::Bhole_binaire::moment_systeme_inf ( )

Calculates the angular momentum of the black hole using the formula at infinity : $J = -\frac{1}{8 \pi} \oint_{\infty} K_j^i m^j {\mathrm d}S_i$ where $\vec{m} = \frac{\partial}{\partial \phi}$ .

Definition at line 173 of file bhole_glob.C.

References Lorene::Tenseur::annule(), Lorene::Tenseur::dec2_dzpuis(), Lorene::Map::get_mg(), Lorene::Mg3d::get_nzone(), Lorene::Map::get_ori_x(), Lorene::Map::get_rot_phi(), Lorene::Tenseur::gradient(), hole1, hole2, Lorene::Tenseur::inc2_dzpuis(), Lorene::Bhole::mp, Lorene::Valeur::mult_cp(), Lorene::Valeur::mult_sp(), Lorene::Valeur::mult_st(), omega, Lorene::Tenseur::set_etat_qcq(), Lorene::Tenseur::set_std_base(), Lorene::Bhole::shift_auto, and Lorene::Cmp::va.

◆ operator()()

const Bhole& Lorene::Bhole_binaire::operator() ( int i ) const

inline

Read only of a component of the system.

i must be equal to 1 or 2.

Definition at line 803 of file bhole.h.

References holes.

◆ operator=()

void Lorene::Bhole_binaire::operator= ( const Bhole_binaire & source )

Affectation operator.

Definition at line 233 of file bhole_binaire.C.

References hole1, hole2, and omega.

◆ set()

Bhole& Lorene::Bhole_binaire::set ( int i )

inline

Read/write of a component of the system.

i must be equal to 1 or 2.

Definition at line 785 of file bhole.h.

References holes.

◆ set_omega()

void Lorene::Bhole_binaire::set_omega ( double ome )

inline

Sets the orbital velocity to ome.

Definition at line 791 of file bhole.h.

References hole1, hole2, omega, and Lorene::Bhole::set_omega().

◆ set_statiques()

void Lorene::Bhole_binaire::set_statiques	(	double	precis,
		double	relax
	)

Initialize the systeme to Misner Lindquist solution, that is solving for N and $Psi$ in the case $\Omega = 0$ .

Parameters

precis	[input] : precision for the convergence (on N ).
relax	[input] : relaxation parameter.

Definition at line 121 of file bhole_coal.C.

References Lorene::diffrelmax(), Lorene::Map::get_mg(), Lorene::Bhole::get_n_auto(), Lorene::Mg3d::get_nzone(), hole1, hole2, init_bhole_binaire(), Lorene::Bhole::mp, set_omega(), solve_lapse(), and solve_psi().

◆ solve_lapse()

void Lorene::Bhole_binaire::solve_lapse	(	double	precis,
		double	relax
	)

Solves the equation for the lapse~:

$\Delta N_a = -\frac{2}{\Psi}\nabla_i \Psi_a \nabla^i N + N \Psi^4 K_{ij}K_a^{ij}$

The fields are the total values excpet those with subscript $_a$ , which are the fields generated by each holes (a = 1, 2).

The boundary conditions are such that N =0 on both horizons. The companion contributions are then obtained.

Parameters

precis	[input] : precision, for the boudary conditions are obtained by iteration.
relax	[input] : relaxation parameter.

Definition at line 87 of file bhole_equations_bin.C.

References Lorene::Bhole::fait_n_comp(), Lorene::flat_scalar_prod(), Lorene::Bhole::grad_n_tot, Lorene::Tenseur::gradient(), hole1, hole2, Lorene::Bhole::n_auto, Lorene::Bhole::n_tot, Lorene::pow(), Lorene::Bhole::psi_auto, Lorene::Bhole::psi_tot, Lorene::Cmp::raccord(), Lorene::Tenseur::set(), Lorene::Cmp::std_base_scal(), Lorene::Bhole::tkij_auto, and Lorene::Bhole::tkij_tot.

◆ solve_phi()

void Lorene::Bhole_binaire::solve_phi	(	double	precision,
		double	relax
	)

Solve the equation for the logarithm of $\Psi$ .

Definition at line 111 of file bhole_solve_phi.C.

References Lorene::diffrelmax(), Lorene::Bhole::fait_psi_comp(), Lorene::flat_scalar_prod(), Lorene::Mg3d::get_angu(), Lorene::Map::get_mg(), Lorene::Bhole::grad_psi_tot, Lorene::Tenseur::gradient(), hole1, hole2, Lorene::Bhole::mp, Lorene::Bhole::psi_auto, Lorene::Cmp::raccord(), Lorene::Bhole::rayon, Lorene::Tenseur::set(), and Lorene::Cmp::std_base_scal().

◆ solve_psi()

void Lorene::Bhole_binaire::solve_psi	(	double	precis,
		double	relax
	)

Solves the equation for the conformal factor~:

$*\Delta \Psi_a = -\frac {\Psi^5}{8} K_{ij}K_a^{ij}$

The fields are the total values excpet those with subscript $_a$ , which are the fields generated by each holes (a = 1, 2).

The boundary conditions are such that $\partial_r \Psi = -\frac{1}{2 r_0} f\left(\theta, \phi\right)$ on both horizons, $r_0$ being the radii of those horizons and f the value of $\Psi$ on the horizons at the preceeding step. The companion contributions are then obtained.

Parameters

precis	[input] : precision, for the boudary conditions are being obtained by iteration.
relax	[input] : relaxation parameter.

Definition at line 137 of file bhole_equations_bin.C.

References Lorene::Bhole::fait_psi_comp(), Lorene::flat_scalar_prod(), Lorene::Mg3d::get_angu(), Lorene::Map::get_mg(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nt(), hole1, hole2, Lorene::Bhole::mp, Lorene::pow(), Lorene::Bhole::psi_auto, Lorene::Bhole::psi_tot, Lorene::Cmp::raccord(), Lorene::Bhole::rayon, Lorene::Valeur::set(), Lorene::Tenseur::set(), Lorene::Cmp::std_base_scal(), Lorene::Bhole::tkij_auto, and Lorene::Bhole::tkij_tot.

◆ solve_shift()

void Lorene::Bhole_binaire::solve_shift	(	double	precis,
		double	relax
	)

Solves the equation for the shift, using the Oohara-Nakarmure scheme~:

$*\Delta \beta_a^i +\frac{1}{3} \nabla^i \left(\nabla_j \beta_a^j\right) = -6A^{ij}\frac{\nabla_i \Psi_a}{\Psi} + 2K^{ij}\nabla_j N_a$

using the current $\Omega$ for the boudary conditions~: $\vec{N} = -\Omega \vec{m}$ on both horizons.

The fields are the total values excpet those with subscript $_a$ , which are the fields generated by each holes (a = 1, 2). The companion contributions are then obtained.

Parameters

precis	[input] : precision for the solver, the boundary conditions and the inversion of the operator being obtained by iteration.
relax	[input] : relaxation parameter.

Definition at line 199 of file bhole_equations_bin.C.

References Lorene::flat_scalar_prod(), Lorene::Tenseur::get_etat(), Lorene::Tenseur::gradient(), hole1, Lorene::Bhole::n_auto, omega, Lorene::Bhole::psi_auto, Lorene::Bhole::psi_tot, Lorene::Bhole::taij_tot, and Lorene::Bhole::tkij_tot.

◆ viriel()

double Lorene::Bhole_binaire::viriel ( ) const

Computes the viriel error, that is the difference between the ADM and the Komar masses, calculated by the asymptotic behaviours of respectively $\Psi$ and N .

Definition at line 96 of file bhole_pseudo_viriel.C.

References Lorene::Map::get_mg(), Lorene::Mg3d::get_nzone(), hole1, hole2, Lorene::Bhole::mp, Lorene::Bhole::n_auto, and Lorene::Bhole::psi_auto.

Member Data Documentation

◆ hole1

Bhole Lorene::Bhole_binaire::hole1

private

Black hole one.

Definition at line 762 of file bhole.h.

◆ hole2

Bhole Lorene::Bhole_binaire::hole2

private

Black hole two.

Definition at line 763 of file bhole.h.

◆ holes

Bhole* Lorene::Bhole_binaire::holes[2]

private

Array on the black holes.

Definition at line 766 of file bhole.h.

◆ omega

double Lorene::Bhole_binaire::omega

private

Position of the axis of rotation.

Angular velocity

Definition at line 769 of file bhole.h.

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

Public Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ Bhole_binaire() [1/2]

◆ Bhole_binaire() [2/2]

◆ ~Bhole_binaire()

Member Function Documentation

◆ adm_systeme()

◆ coal()

◆ distance_propre()

◆ fait_decouple()

◆ fait_tkij()

◆ get_omega()

◆ init_bhole_binaire()

◆ init_phi()

◆ komar_systeme()

◆ moment_systeme_hor()

◆ moment_systeme_inf()

◆ operator()()

◆ operator=()

◆ set()

◆ set_omega()

◆ set_statiques()

◆ solve_lapse()

◆ solve_phi()

◆ solve_psi()

◆ solve_shift()

◆ viriel()

Member Data Documentation

◆ hole1

◆ hole2

◆ holes

◆ omega