Bhole_binaire Class Reference
[Stars and black holes]

Binary black holes system. More...

#include <bhole.h>

Public Member Functions
	Bhole_binaire (Map_af &mp1, Map_af &mp2)
	Standard constructor.
	Bhole_binaire (const Bhole_binaire &)
	Copy.
	~Bhole_binaire ()
	Destructor.
void	operator= (const Bhole_binaire &)
	Affectation operator.
Bhole &	set (int i)
	Read/write of a component of the system.
void	set_omega (double ome)
	Sets the orbital velocity to `ome`.
void	set_pos_axe (double)
const Bhole &	operator() (int i) const
	Read only of a component of the system.
double	get_omega () const
	Returns the angular velocity.
void	init_bhole_binaire ()
	Initialisation of the system.
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 .
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).
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).
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.
void	fait_tkij ()
	Calculation af the extrinsic curvature tensor.
void	fait_decouple ()
	Calculates {tt decouple} which is used to obtain `tkij_auto` by the formula : `tkij_auto` = `decouple` * `tkij_tot` .
void	set_statiques (double precis, double relax)
	Initialize the systeme to Misner Lindquist solution, that is solving for N and in the case $\Omega = 0$ .
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.
double	adm_systeme () const
	Calculates the ADM mass of the system using : $M = -\frac{1}{2 \pi} \oint_{\infty} \nabla^i \Psi {\mathrm d} S_i$ .
double	komar_systeme () const
	Calculates the Komar mass of the system using : $M = \frac{1}{4 \pi} \oint_{\infty} \nabla^i N {\mathrm d} S_i$ .
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 $\vec{m} = \frac{\partial}{\partial \phi}$ .
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 .
double	distance_propre (const int nr=65) const
	Calculation of the proper distance between the two spheres of inversion, along the x axis.
Tbl	linear_momentum_systeme_inf () const
void	solve_phi (double precision, double relax)
	Solve the equation for the logarithm of $\Psi$ .
void	init_phi ()
	Initiates the system for a resolution using the logarithm of $\Psi$ .
Private Attributes
Bhole	hole1
	Black hole one.
Bhole	hole2
	Black hole two.
Bhole *	holes [2]
	Array on the black holes.
double	pos_axe
double	omega
	Position of the axis of rotation.

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 752 of file bhole.h.

Constructor & Destructor Documentation

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

Standard constructor.

Definition at line 207 of file bhole_binaire.C.

References hole1, hole2, and holes.

Bhole_binaire::Bhole_binaire ( const Bhole_binaire & source )

Copy.

Definition at line 215 of file bhole_binaire.C.

References hole1, hole2, and holes.

Bhole_binaire::~Bhole_binaire ( )

Destructor.

Definition at line 222 of file bhole_binaire.C.

Member Function Documentation

double 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 82 of file bhole_glob.C.

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

void 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 153 of file bhole_coal.C.

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

double 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 258 of file bhole_glob.C.

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

void 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 310 of file bhole_kij.C.

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

void 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 84 of file bhole_kij.C.

References Map::convert_absolute(), Cmp::dec2_dzpuis(), Tenseur::dec2_dzpuis(), Bhole::decouple, Bhole::fait_taij_auto(), Map_af::get_alpha(), Map_af::get_beta(), Tenseur::get_etat(), Map::get_mg(), Tenseur::get_mp(), Mg3d::get_np(), Mg3d::get_nr(), Mg3d::get_nt(), Mg3d::get_nzone(), Map::get_rot_phi(), hole1, hole2, Cmp::import_asymy(), Cmp::import_symy(), Cmp::inc2_dzpuis(), Tenseur::inc2_dzpuis(), Bhole::mp, Bhole::n_tot, Cmp::raccord(), Cmp::set(), Tenseur::set(), Tenseur::set_etat_qcq(), Tenseur::set_etat_zero(), Tenseur::set_std_base(), Bhole::taij_auto, Bhole::taij_comp, Bhole::taij_tot, Bhole::tkij_auto, Bhole::tkij_tot, Cmp::val_point(), Map::xa, Map::ya, and Map::za.

double Bhole_binaire::get_omega ( ) const [inline]

Returns the angular velocity.

Definition at line 803 of file bhole.h.

References omega.

void 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 235 of file bhole_binaire.C.

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

void Bhole_binaire::init_phi ( )

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

Definition at line 151 of file bhole_solve_phi.C.

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

double 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 93 of file bhole_glob.C.

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

double 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 104 of file bhole_glob.C.

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

double 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 166 of file bhole_glob.C.

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

const Bhole& 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 798 of file bhole.h.

References holes.

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

Affectation operator.

Definition at line 226 of file bhole_binaire.C.

References hole1, hole2, and omega.

Bhole& 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 780 of file bhole.h.

References holes.

void Bhole_binaire::set_omega ( double ome ) [inline]

Sets the orbital velocity to ome.

Definition at line 786 of file bhole.h.

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

void 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 114 of file bhole_coal.C.

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

void 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 80 of file bhole_equations_bin.C.

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

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

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

Definition at line 104 of file bhole_solve_phi.C.

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

void 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 130 of file bhole_equations_bin.C.

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

void 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 192 of file bhole_equations_bin.C.

References Cmp::filtre(), flat_scalar_prod(), Mg3d::get_angu(), Tenseur::get_etat(), Map::get_mg(), Tenseur::get_mp(), Mg3d::get_np(), Mg3d::get_nt(), Map::get_rot_phi(), Tenseur::gradient(), hole1, hole2, Bhole::mp, Bhole::n_auto, omega, Bhole::omega_local, Bhole::psi_auto, Bhole::psi_tot, Cmp::raccord(), Bhole::regul, Coord::set(), Tenseur::set(), Tenseur::set_etat_qcq(), Tenseur::set_etat_zero(), Cmp::set_etat_zero(), Tenseur::set_std_base(), Bhole::shift_auto, Mg3d::std_base_vect_cart(), Bhole::taij_tot, Bhole::tkij_tot, Map::x, Map::xa, Map::y, and Map::ya.

double 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 89 of file bhole_pseudo_viriel.C.

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