Black_hole Class Reference
[Stars and black holes]

Base class for black holes. More...

#include <blackhole.h>

Inheritance diagram for Black_hole:

Public Member Functions
	Black_hole (Map &mp_i, bool Kerr_schild, double massbh)
	Standard constructor.
	Black_hole (const Black_hole &)
	Copy constructor.
	Black_hole (Map &mp_i, FILE *fich)
	Constructor from a file (see `sauve(FILE*)` ).
virtual	~Black_hole ()
	Destructor.
void	operator= (const Black_hole &)
	Assignment to another `Black_hole`.
Map &	set_mp ()
	Read/write of the mapping.
double &	set_mass_bh ()
	Read/write of the gravitational mass of BH [{ m}].
const Map &	get_mp () const
	Returns the mapping.
bool	is_kerrschild () const
	Returns `true` for a Kerr-Schild background, `false` for a Conformally flat one.
double	get_mass_bh () const
	Returns the gravitational mass of BH [{ m}].
const Scalar &	get_lapconf () const
	Returns lapconf generated by the black hole.
const Scalar &	get_lapconf_rs () const
	Returns the part of lapconf from the numerical computation.
const Scalar &	get_lapse () const
	Returns the lapse function generated by the black hole.
const Vector &	get_shift () const
	Returns the shift vector generated by the black hole.
const Vector &	get_shift_rs () const
	Returns the part of the shift vector from the numerical computation.
const Scalar &	get_confo () const
	Returns the conformal factor generated by the black hole.
virtual void	sauve (FILE *) const
	Save in a file.
virtual double	mass_irr () const
	Irreducible mass of the black hole.
virtual double	mass_adm () const
	ADM mass.
virtual double	mass_kom () const
	Komar mass.
virtual double	rad_ah () const
	Radius of the apparent horizon.
double	spin_am_bh (bool bclapconf_nd, bool bclapconf_fs, const Tbl &xi_i, const double &phi_i, const double &theta_i, const int &nrk_phi, const int &nrk_theta) const
	Spin angular momentum.
const Tbl &	angu_mom_bh () const
	Total angular momentum.
const Valeur	bc_lapconf (bool neumann, bool first) const
	Boundary condition on the apparent horizon of the black hole for the lapse function: 2-D `Valeur`.
const Valeur	bc_shift_x (double omega_r) const
	Boundary condition on the apparent horizon of the black hole for the shift vector of the direction: 2-D `Valeur`.
const Valeur	bc_shift_y (double omega_r) const
	Boundary condition on the apparent horizon of the black hole for the shift vector of the direction: 2-D `Valeur`.
const Valeur	bc_shift_z () const
	Boundary condition on the apparent horizon of the black hole for the shift vector of the direction: 2-D `Valeur`.
const Valeur	bc_confo () const
	Boundary condition on the apparent horizon of the black hole for the conformal factor: 2-D `Valeur`.
void	extr_curv_bh ()
	Computes `taij` , `taij_quad` from `shift` , `lapse` , `confo` .
void	equilibrium_spher (bool neumann, bool first, double spin_omega, double precis=1.e-14, double precis_shift=1.e-8)
	Computes a spherical, static black-hole by giving boundary conditions on the apparent horizon.
void	static_bh (bool neumann, bool first)
	Sets the metric quantities to a spherical, static blach-hole analytic solution.
double	rah_iso (bool neumann, bool first) const
	Computes a radius of apparent horizon (excised surface) in isotropic coordinates.
const Scalar	r_coord (bool neumann, bool first) const
	Expresses the areal radial coordinate by that in spatially isotropic coordinates.
Tbl	runge_kutta_phi_bh (const Tbl &xi_i, const double &phi_i, const int &nrk) const
	Compute a forth-order Runge-Kutta integration to the phi direction for the solution of the Killing vectors on the equator.
Tbl	runge_kutta_theta_bh (const Tbl &xi_i, const double &theta_i, const double &phi, const int &nrk) const
	Compute a forth-order Runge-Kutta integration to the theta direction for the solution of the Killing vectors.
Vector	killing_vect_bh (const Tbl &xi_i, const double &phi_i, const double &theta_i, const int &nrk_phi, const int &nrk_theta) const
	Compute the Killing vector of a black hole normalized so that its affine length is 2 M_PI.
Protected Member Functions
virtual void	del_deriv () const
	Deletes all the derived quantities.
void	set_der_0x0 () const
	Sets to `0x0` all the pointers on derived quantities.
virtual ostream &	operator>> (ostream &) const
	Operator >> (virtual function called by the operator <<).
Protected Attributes
Map &	mp
	Mapping associated with the black hole.
bool	kerrschild
	`true` for a Kerr-Schild background, `false` for a conformally flat background
double	mass_bh
	Gravitational mass of BH.
Scalar	lapconf
	A function (lapse function * conformal factor) lapconf generated by the black hole.
Scalar	lapconf_rs
	Part of lapconf from the numerical computation.
Scalar	lapconf_bh
	Part of lapconf from the analytic background.
Scalar	lapse
	Lapse function generated by the black hole.
Vector	shift
	Shift vector generated by the black hole.
Vector	shift_rs
	Part of the shift vector from the numerical computation.
Vector	shift_bh
	Part of the shift vector from the analytic background.
Scalar	confo
	Conformal factor generated by the black hole.
Sym_tensor	taij
	Trace of the extrinsic curvature.
Sym_tensor	taij_rs
	Part of the extrinsic curvature tensor.
Scalar	taij_quad
	Part of the scalar $\eta_{ik} \eta_{jl} A^{ij} A^{kl}$ generated by $A_{ij}$ .
Scalar	taij_quad_rs
	Part of the scalar.
Metric_flat	flat
	Flat metric defined on the mapping (Spherical components with respect to the mapping of the black hole ).
double *	p_mass_irr
	Conformal metric $\tilde \gamma_{ij}$ .
double *	p_mass_adm
	Irreducible mass of the black hole.
double *	p_mass_kom
	ADM mass.
double *	p_rad_ah
	Komar mass.
double *	p_spin_am_bh
	Radius of the apparent horizon.
Tbl *	p_angu_mom_bh
	Spin angular momentum.
Friends
ostream &	operator<< (ostream &, const Black_hole &)
	Display.

Detailed Description

Base class for black holes.

()

According to the 3+1 formalism, the spacetime metric is written

$ds^2 = - \alpha^2 dt^2 + \gamma_{ij} ( dx^i + \beta^i dt ) ( dx^j + \beta^j dt )$

where $\gamma_{ij}$ is the spatial metric.

Definition at line 70 of file blackhole.h.

Constructor & Destructor Documentation

Black_hole::Black_hole	(	Map &	mp_i,
		bool	Kerr_schild,
		double	massbh
	)

Standard constructor.

Parameters:

mp_i

Mapping on which the black hole will be defined

Definition at line 67 of file blackhole.C.

References confo, lapconf, lapconf_bh, lapconf_rs, lapse, set_der_0x0(), Tensor::set_etat_zero(), shift, shift_bh, shift_rs, Scalar::std_spectral_base(), taij, taij_quad, taij_quad_rs, and taij_rs.

Black_hole::Black_hole ( const Black_hole & bh )

Copy constructor.

Definition at line 114 of file blackhole.C.

References set_der_0x0().

Black_hole::Black_hole	(	Map &	mp_i,
		FILE *	fich
	)

Constructor from a file (see sauve(FILE*) ).

Parameters:

	mp_i	Mapping on which the black hole will be defined
	fich	input file (must have been created by the function `sauve` )

Definition at line 138 of file blackhole.C.

References confo, kerrschild, lapconf, lapconf_bh, lapconf_rs, lapse, mass_bh, set_der_0x0(), Tensor::set_etat_zero(), shift, shift_bh, shift_rs, Vector::std_spectral_base(), Scalar::std_spectral_base(), taij, taij_quad, taij_quad_rs, and taij_rs.

Black_hole::~Black_hole ( ) [virtual]

Destructor.

Definition at line 190 of file blackhole.C.

References del_deriv().

Member Function Documentation

const Tbl & Black_hole::angu_mom_bh ( ) const

Total angular momentum.

Returns:: 1-D { Tbl} of size 3, according to \ { angu()(0)} = $J^x$, \ { angu()(1)} = $J^y$, \ { angu()(2)} = $J^z$.

Definition at line 630 of file blackhole_global.C.

References Tensor::annule_domain(), Tbl::annule_hard(), Map::cosp, Map::cost, Scalar::dec_dzpuis(), Map::get_bvect_cart(), Map_af::integrale_surface(), mp, p_angu_mom_bh, Scalar::raccord(), Tbl::set(), Tensor::set(), Map::sinp, Map::sint, Scalar::std_spectral_base(), taij, Map::val_r(), Map::x, Map::y, and Map::z.

const Valeur Black_hole::bc_confo ( ) const

Boundary condition on the apparent horizon of the black hole for the conformal factor: 2-D Valeur.

Definition at line 406 of file blackhole_bc.C.

References confo, Map::cosp, Map::cost, Scalar::dsdr(), Mg3d::get_angu(), Scalar::get_dzpuis(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nt(), kerrschild, lapconf, mass_bh, mp, pow(), Map::r, Valeur::set(), Scalar::set_dzpuis(), shift, Map::sinp, Map::sint, Valeur::std_base_scal(), Scalar::std_spectral_base(), and Scalar::val_grid_point().

const Valeur Black_hole::bc_lapconf	(	bool	neumann,
		bool	first
	)			const

Boundary condition on the apparent horizon of the black hole for the lapse function: 2-D Valeur.

Definition at line 67 of file blackhole_bc.C.

References Mg3d::get_angu(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nt(), kerrschild, lapconf, mp, Map::r, Valeur::set(), Valeur::std_base_scal(), Scalar::std_spectral_base(), and Scalar::val_grid_point().

const Valeur Black_hole::bc_shift_x ( double omega_r ) const

Boundary condition on the apparent horizon of the black hole for the shift vector of the direction: 2-D Valeur.

Definition at line 207 of file blackhole_bc.C.

References Valeur::base, confo, Map::cosp, Mg3d::get_angu(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nt(), kerrschild, lapconf, mp, pow(), Map::r, Valeur::set(), Map::sint, Mg3d::std_base_vect_cart(), Scalar::std_spectral_base(), Scalar::val_grid_point(), and Map::y.

const Valeur Black_hole::bc_shift_y ( double omega_r ) const

Boundary condition on the apparent horizon of the black hole for the shift vector of the direction: 2-D Valeur.

Definition at line 277 of file blackhole_bc.C.

References Valeur::base, confo, Mg3d::get_angu(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nt(), kerrschild, lapconf, mp, pow(), Map::r, Valeur::set(), Map::sinp, Map::sint, Mg3d::std_base_vect_cart(), Scalar::std_spectral_base(), Scalar::val_grid_point(), and Map::x.

const Valeur Black_hole::bc_shift_z ( ) const

Boundary condition on the apparent horizon of the black hole for the shift vector of the direction: 2-D Valeur.

Reimplemented in Hole_bhns.

Definition at line 346 of file blackhole_bc.C.

References Valeur::base, confo, Map::cost, Mg3d::get_angu(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nt(), kerrschild, lapconf, mp, pow(), Map::r, Valeur::set(), Mg3d::std_base_vect_cart(), Scalar::std_spectral_base(), and Scalar::val_grid_point().

void Black_hole::del_deriv ( ) const [protected, virtual]

Deletes all the derived quantities.

Reimplemented in Hole_bhns.

Definition at line 201 of file blackhole.C.

References p_angu_mom_bh, p_mass_adm, p_mass_irr, p_mass_kom, p_rad_ah, p_spin_am_bh, and set_der_0x0().

void Black_hole::equilibrium_spher	(	bool	neumann,
		bool	first,
		double	spin_omega,
		double	precis = `1.e-14`,
		double	precis_shift = `1.e-8`
	)

Computes a spherical, static black-hole by giving boundary conditions on the apparent horizon.

Parameters:

	neumann	[input] `true` for the neumann bc, `false` for the dirichlet one for the lapse
	first	[input] `true` for the first type of bc, `false` for the second type
	spin_omega	[input] spin parameter of the black hole
	precis	[input] threshold in the relative difference of a metric quantity between two succesive steps to stop the iterative procedure (default value: 1.e-14)
	precis_shift	[input] threshold in the relative difference of the shift vector between two succesive steps to stop the iterative procedure (default value: 1.e-8)

Definition at line 66 of file blackhole_eq_spher.C.

References Tensor::annule_domain(), bc_confo(), bc_lapconf(), bc_shift_x(), bc_shift_y(), bc_shift_z(), confo, contract(), Map::cosp, Map::cost, del_deriv(), Scalar::deriv(), diffrel(), extr_curv_bh(), Mg3d::get_angu(), Map::get_bvect_cart(), Scalar::get_dzpuis(), Map::get_mg(), Mg3d::get_nzone(), Scalar::inc_dzpuis(), kerrschild, lapconf, lapconf_bh, lapconf_rs, lapse, mass_adm(), mass_bh, mass_irr(), mass_kom(), mp, Scalar::poisson_dirichlet(), Scalar::poisson_neumann(), pow(), Map::r, r_coord(), Scalar::raccord(), rad_ah(), Tbl::set(), Vector::set(), Tensor::set(), Scalar::set_dzpuis(), Scalar::set_etat_qcq(), Tensor::set_etat_zero(), shift, shift_bh, shift_rs, Map::sinp, Map::sint, sqrt(), Vector::std_spectral_base(), Scalar::std_spectral_base(), taij, and taij_quad.

void Black_hole::extr_curv_bh ( )

Computes taij , taij_quad from shift , lapse , confo .

Definition at line 58 of file blackhole_extr_curv.C.

References Tensor::annule_domain(), Metric_flat::con(), confo, Map::cosp, Map::cost, Metric_flat::cov(), flat, Map::get_bvect_cart(), Scalar::inc_dzpuis(), kerrschild, lapconf, mass_bh, mp, pow(), Map::r, Scalar::raccord(), Tensor::set(), Tensor::set_etat_qcq(), shift, shift_rs, Map::sinp, Map::sint, sqrt(), Tensor::std_spectral_base(), Scalar::std_spectral_base(), taij, taij_quad, taij_quad_rs, and taij_rs.

const Scalar& Black_hole::get_confo ( ) const [inline]

Returns the conformal factor generated by the black hole.

Definition at line 237 of file blackhole.h.

References confo.

const Scalar& Black_hole::get_lapconf ( ) const [inline]

Returns lapconf generated by the black hole.

Definition at line 220 of file blackhole.h.

References lapconf.

const Scalar& Black_hole::get_lapconf_rs ( ) const [inline]

Returns the part of lapconf from the numerical computation.

Definition at line 223 of file blackhole.h.

References lapconf_rs.

const Scalar& Black_hole::get_lapse ( ) const [inline]

Returns the lapse function generated by the black hole.

Definition at line 226 of file blackhole.h.

References lapse.

double Black_hole::get_mass_bh ( ) const [inline]

Returns the gravitational mass of BH [{ m}].

Definition at line 217 of file blackhole.h.

References mass_bh.

const Map& Black_hole::get_mp ( ) const [inline]

Returns the mapping.

Definition at line 209 of file blackhole.h.

References mp.

const Vector& Black_hole::get_shift ( ) const [inline]

Returns the shift vector generated by the black hole.

Definition at line 229 of file blackhole.h.

References shift.

const Vector& Black_hole::get_shift_rs ( ) const [inline]

Returns the part of the shift vector from the numerical computation.

Definition at line 234 of file blackhole.h.

References shift_rs.

bool Black_hole::is_kerrschild ( ) const [inline]

Returns true for a Kerr-Schild background, false for a Conformally flat one.

Definition at line 214 of file blackhole.h.

References kerrschild.

Vector Black_hole::killing_vect_bh	(	const Tbl &	xi_i,
		const double &	phi_i,
		const double &	theta_i,
		const int &	nrk_phi,
		const int &	nrk_theta
	)			const

Compute the Killing vector of a black hole normalized so that its affine length is 2 M_PI.

Parameters:

	xi_i	[input] initial set of the Killing vectors at the starting point on the equator
	phi_i	[input] initial phi coordinate at which the integration starts
	theta_i	[input] initial theta coordinate at which the integration starts
	nrk_phi	[input] number of the Runge-Kutta integration between two successive collocation points for the phi direction
	nrk_theta	[input] number of the Runge-Kutta integration between two successive collocation points for the theta direction

Definition at line 62 of file blackhole_killing.C.

References confo, Map::get_bvect_spher(), Tbl::get_dim(), Map::get_mg(), Tbl::get_ndim(), Mg3d::get_nr(), kerrschild, mp, pow(), Map::r, rad_ah(), runge_kutta_phi_bh(), runge_kutta_theta_bh(), Tbl::set(), Tbl::set_etat_qcq(), Scalar::set_grid_point(), Map::sint, Scalar::std_spectral_base(), Scalar::val_grid_point(), and Scalar::val_point().

double Black_hole::mass_adm ( ) const [virtual]

ADM mass.

Definition at line 99 of file blackhole_global.C.

References Tensor::annule_domain(), confo, Map::cosp, Map::cost, Scalar::dec_dzpuis(), Scalar::dsdr(), Map::get_bvect_cart(), Scalar::integrale(), Map_af::integrale_surface(), Map_af::integrale_surface_infini(), kerrschild, lapconf, mp, p_mass_adm, pow(), Map::r, Scalar::raccord(), Tensor::set(), Scalar::set_etat_qcq(), shift, Map::sinp, Map::sint, Scalar::std_spectral_base(), taij_quad, and Map::val_r().

double Black_hole::mass_irr ( ) const [virtual]

Irreducible mass of the black hole.

Definition at line 65 of file blackhole_global.C.

References Tensor::annule_domain(), confo, Map_af::integrale_surface(), mp, p_mass_irr, pow(), Scalar::raccord(), sqrt(), Scalar::std_spectral_base(), and Map::val_r().

double Black_hole::mass_kom ( ) const [virtual]

Komar mass.

Definition at line 285 of file blackhole_global.C.

References Tensor::annule_domain(), confo, Map::cosp, Map::cost, Scalar::dec_dzpuis(), Scalar::deriv(), Scalar::dsdr(), Map::get_bvect_cart(), Scalar::integrale(), Map_af::integrale_surface(), Map_af::integrale_surface_infini(), kerrschild, lapconf, mp, p_mass_kom, pow(), Map::r, Scalar::raccord(), Tensor::set(), Scalar::set_etat_qcq(), Map::sinp, Map::sint, Scalar::std_spectral_base(), taij_quad, and Map::val_r().

void Black_hole::operator= ( const Black_hole & bh )

Assignment to another Black_hole.

Reimplemented in Hole_bhns.

Definition at line 232 of file blackhole.C.

References confo, del_deriv(), flat, kerrschild, lapconf, lapconf_bh, lapconf_rs, lapse, mass_bh, mp, shift, shift_bh, shift_rs, taij, taij_quad, taij_quad_rs, and taij_rs.

ostream & Black_hole::operator>> ( ostream & ost ) const [protected, virtual]

Operator >> (virtual function called by the operator <<).

Reimplemented in Hole_bhns.

Definition at line 283 of file blackhole.C.

References confo, Map::get_mg(), Mg3d::get_nt(), kerrschild, lapconf, lapse, mass_adm(), mass_bh, mass_irr(), mass_kom(), mp, shift, and Scalar::val_grid_point().

const Scalar Black_hole::r_coord	(	bool	neumann,
		bool	first
	)			const

Expresses the areal radial coordinate by that in spatially isotropic coordinates.

Parameters:

	neumann	[input] `true` for the neumann bc, `false` for the dirichlet one for the lapse
	first	[input] `true` for the first type of bc, `false` for the second type

Definition at line 62 of file blackhole_r_coord.C.

References Tensor::annule_domain(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nr(), Mg3d::get_nt(), Mg3d::get_nzone(), log(), mass_bh, mp, Map::r, Scalar::raccord(), Scalar::set_grid_point(), sqrt(), Scalar::std_spectral_base(), and Scalar::val_grid_point().

double Black_hole::rad_ah ( ) const [virtual]

Radius of the apparent horizon.

Definition at line 501 of file blackhole_global.C.

References mp, p_rad_ah, Map::r, Scalar::std_spectral_base(), and Scalar::val_grid_point().

double Black_hole::rah_iso	(	bool	neumann,
		bool	first
	)			const

Computes a radius of apparent horizon (excised surface) in isotropic coordinates.

Parameters:

	neumann	[input] `true` for the neumann bc, `false` for the dirichlet one for the lapse
	first	[input] `true` for the first type of bc, `false` for the second type

Definition at line 65 of file blackhole_rah_iso.C.

References exp(), and sqrt().

Tbl Black_hole::runge_kutta_phi_bh	(	const Tbl &	xi_i,
		const double &	phi_i,
		const int &	nrk
	)			const

Compute a forth-order Runge-Kutta integration to the phi direction for the solution of the Killing vectors on the equator.

Parameters:

	xi_i	[input] initial set of the Killing vectors at the starting point on the equator
	phi_i	[input] initial phi coordinate at which the integration starts
	nrk	[input] number of the Runge-Kutta integration between two successive collocation points

Definition at line 63 of file blackhole_rk_phi.C.

References confo, Scalar::dsdt(), Map::get_mg(), Mg3d::get_np(), kerrschild, Scalar::lapang(), mp, rad_ah(), Tbl::set(), Tbl::set_etat_qcq(), Scalar::std_spectral_base(), and Scalar::val_point().

Tbl Black_hole::runge_kutta_theta_bh	(	const Tbl &	xi_i,
		const double &	theta_i,
		const double &	phi,
		const int &	nrk
	)			const

Compute a forth-order Runge-Kutta integration to the theta direction for the solution of the Killing vectors.

Parameters:

	xi_i	[input] initial set of the Killing vectors at the starting point on the equator
	theta_i	[input] initial theta coordinate at which the integration starts
	phi	[input] fixed phi coordinate during integration to the theta direction
	nrk	[input] number of the Runge-Kutta integration between two successive collocation points

Definition at line 63 of file blackhole_rk_theta.C.

References confo, Map::get_mg(), Mg3d::get_nt(), kerrschild, Scalar::lapang(), mp, rad_ah(), Tbl::set(), Tbl::set_etat_qcq(), Scalar::std_spectral_base(), Scalar::stdsdp(), and Scalar::val_point().

void Black_hole::sauve ( FILE * fich ) const [virtual]

Save in a file.

Reimplemented in Hole_bhns.

Definition at line 263 of file blackhole.C.

References confo, kerrschild, lapconf_rs, mass_bh, Tensor::sauve(), Scalar::sauve(), and shift_rs.

void Black_hole::set_der_0x0 ( ) const [protected]

Sets to 0x0 all the pointers on derived quantities.

Reimplemented in Hole_bhns.

Definition at line 214 of file blackhole.C.

References p_angu_mom_bh, p_mass_adm, p_mass_irr, p_mass_kom, p_rad_ah, and p_spin_am_bh.

double& Black_hole::set_mass_bh ( ) [inline]

Read/write of the gravitational mass of BH [{ m}].

Definition at line 203 of file blackhole.h.

References mass_bh.

Map& Black_hole::set_mp ( ) [inline]

Read/write of the mapping.

Definition at line 200 of file blackhole.h.

References mp.

double Black_hole::spin_am_bh	(	bool	bclapconf_nd,
		bool	bclapconf_fs,
		const Tbl &	xi_i,
		const double &	phi_i,
		const double &	theta_i,
		const int &	nrk_phi,
		const int &	nrk_theta
	)			const

Spin angular momentum.

Definition at line 524 of file blackhole_global.C.

References Tensor::annule_domain(), confo, Map::cosp, Map::cost, Scalar::dec_dzpuis(), Map::get_bvect_cart(), Map::get_bvect_spher(), Map_af::integrale_surface(), kerrschild, killing_vect_bh(), mp, p_spin_am_bh, pow(), Scalar::raccord(), Tensor::set(), Map::sinp, Map::sint, Scalar::std_spectral_base(), taij, and Map::val_r().

void Black_hole::static_bh	(	bool	neumann,
		bool	first
	)

Sets the metric quantities to a spherical, static blach-hole analytic solution.

Parameters:

	neumann	[input] `true` for the neumann bc, `false` for the dirichlet one for the lapse
	first	[input] `true` for the first type of bc, `false` for the second type