Kerr spacetime in Quasi-Isotropic coordinates (under development). More...

#include <compobj.h>

Inheritance diagram for Lorene::Kerr_QI:

Public Member Functions
	Kerr_QI (Map &mp_i, double mass, double a_over_m)
	Standard constructor.
	Kerr_QI (const Kerr_QI &)
	Copy constructor.
	Kerr_QI (Map &mp_i, FILE *fich)
	Constructor from a file (see `sauve(FILE*)` ).
virtual	~Kerr_QI ()
	Destructor.
void	operator= (const Kerr_QI &)
	Assignment to another `Kerr_QI`.
virtual void	sauve (FILE *) const
	Save in a file.
const Scalar &	get_bbb () const
	Returns the metric factor B.
const Scalar &	get_a_car () const
	Returns the square of the metric factor A.
const Scalar &	get_b_car () const
	Returns the square of the metric factor B.
const Scalar &	get_nphi () const
	Returns the metric coefficient $N^\varphi$ .
const Scalar &	get_ak_car () const
	Returns the scalar $A^2 K_{ij} K^{ij}$ .
void	gyoto_data (const char *file_name) const
	Save in a file for GYOTO.
virtual double	angu_mom () const
	Angular momentum.
virtual double	r_isco (int lmin, ostream *ost=0x0) const
	Coordinate r of the innermost stable circular orbit (ISCO).
virtual double	f_isco (int lmin) const
	Orbital frequency at the innermost stable circular orbit (ISCO).
virtual double	espec_isco (int lmin) const
	Energy of a particle at the ISCO.
virtual double	lspec_isco (int lmin) const
	Angular momentum of a particle at the ISCO.
virtual double	r_mb (int lmin, ostream *ost=0x0) const
	Coordinate r of the marginally bound circular orbit (R_mb).
virtual void	update_metric ()
	Updates the 3-metric $\gamma_{ij}$ from A and B and the shift vector $\beta^i$ from $N^\phi$ .
virtual void	extrinsic_curvature ()
	Computes the extrinsic curvature and `ak_car` from `nphi` , `nn` and `b_car` .
Map &	set_mp ()
	Read/write of the mapping.
const Map &	get_mp () const
	Returns the mapping.
const Scalar &	get_nn () const
	Returns the lapse function N .
const Vector &	get_beta () const
	Returns the shift vector $\beta^i$ .
const Metric &	get_gamma () const
	Returns the 3-metric $\gamma_{ij}$ .
const Scalar &	get_ener_euler () const
	Returns the total energy density E in the Eulerian frame.
const Vector &	get_mom_euler () const
	Returns the total 3-momentum density in the Eulerian frame.
const Sym_tensor &	get_stress_euler () const
	Returns the stress tensor $S_{ij}$ with respect to the Eulerian observer.
const Sym_tensor &	get_kk () const
	Returns the extrinsic curvature tensor $K_{ij}$ .
virtual double	adm_mass () const
	ADM mass (computed as a surface integral at spatial infinity).

Protected Member Functions
virtual void	del_deriv () const
	Deletes all the derived quantities.
virtual 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
double	mm
	mass parameter
double	aa
	angular momentum parameter
Scalar	a_car
	Square of the metric factor A.
Scalar	bbb
	Metric factor B.
Scalar	b_car
	Square of the metric factor B.
Scalar	nphi
	Metric coefficient $N^\varphi$ .
Scalar	ak_car
	Scalar $A^2 K_{ij} K^{ij}$ .
double *	p_angu_mom
	Angular momentum.
double *	p_r_isco
	Coordinate r of the ISCO.
double *	p_f_isco
	Orbital frequency of the ISCO.
double *	p_espec_isco
	Specific energy of a particle at the ISCO.
double *	p_lspec_isco
	Specific angular momentum of a particle at the ISCO.
double *	p_r_mb
	Coordinate r of the marginally bound orbit.
Map &	mp
	Mapping describing the coordinate system (r,theta,phi).
Scalar	nn
	Lapse function N .
Vector	beta
	Shift vector $\beta^i$ .
Metric	gamma
	3-metric $\gamma_{ij}$
Scalar	ener_euler
	Total energy density E in the Eulerian frame.
Vector	mom_euler
	Total 3-momentum density in the Eulerian frame.
Sym_tensor	stress_euler
	Stress tensor $S_{ij}$ with respect to the Eulerian observer.
Sym_tensor	kk
	Extrinsic curvature tensor $K_{ij}$ .
double *	p_adm_mass
	ADM mass.

Detailed Description

Kerr spacetime in Quasi-Isotropic coordinates (under development).

()

The metric is expressed in Quasi-Isotropic (QI) coordinates :

$\‍[ ds^2 = - N^2 dt^2 + A^2 (dr^2 + r^2 d\theta^2) + B^2 r^2 \sin^2\theta (d\varphi - N^\varphi dt)^2 \‍]$

Definition at line 816 of file compobj.h.

Constructor & Destructor Documentation

◆ Kerr_QI() [1/3]

Lorene::Kerr_QI::Kerr_QI	(	Map &	mp_i,
		double	mass,
		double	a_over_m )

Standard constructor.

Parameters

mp_i	Mapping on which the star is contructed
mass	Black hole mass M
a_over_m	Black hole reduced angular momentum a/M (dimensionless)

Definition at line 65 of file kerr_QI.C.

References Lorene::Compobj_QI::a_car, aa, Lorene::Compobj_QI::b_car, Lorene::Compobj_QI::bbb, Lorene::Compobj_QI::Compobj_QI(), Lorene::cost, Lorene::Map(), mm, Lorene::Compobj::mp, Lorene::Compobj::nn, Lorene::Compobj_QI::nphi, Lorene::r, set_der_0x0(), Lorene::sint, and Lorene::sqrt().

◆ Kerr_QI() [2/3]

Lorene::Kerr_QI::Kerr_QI ( const Kerr_QI & other )

Copy constructor.

Definition at line 120 of file kerr_QI.C.

References aa, Lorene::Compobj_QI::Compobj_QI(), Kerr_QI(), mm, and set_der_0x0().

◆ Kerr_QI() [3/3]

Lorene::Kerr_QI::Kerr_QI	(	Map &	mp_i,
		FILE *	fich )

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

Parameters

mp_i	Mapping on which the star is constructed
fich	input file (must have been created by the function `Kerr_QI::sauve` )

Definition at line 132 of file kerr_QI.C.

References Lorene::Compobj_QI::Compobj_QI(), Lorene::Map(), and set_der_0x0().

◆ ~Kerr_QI()

Lorene::Kerr_QI::~Kerr_QI ( )

virtual

Destructor.

Definition at line 148 of file kerr_QI.C.

References del_deriv().

Member Function Documentation

◆ adm_mass()

double Lorene::Compobj::adm_mass ( ) const

virtualinherited

ADM mass (computed as a surface integral at spatial infinity).

Definition at line 310 of file compobj.C.

References Lorene::Tensor::derive_con(), Lorene::Tensor_sym::derive_con(), Lorene::Vector::flux(), gamma, Lorene::Tensor::get_triad(), mp, p_adm_mass, Lorene::Tensor::trace(), and Lorene::Tensor::up().

◆ angu_mom()

double Lorene::Compobj_QI::angu_mom ( ) const

virtualinherited

Angular momentum.

Reimplemented in Lorene::Star_QI.

Definition at line 90 of file compobj_QI_global.C.

References p_angu_mom.

◆ del_deriv()

void Lorene::Kerr_QI::del_deriv ( ) const

protectedvirtual

Deletes all the derived quantities.

Reimplemented from Lorene::Compobj_QI.

Definition at line 159 of file kerr_QI.C.

References Lorene::Compobj_QI::del_deriv(), and set_der_0x0().

◆ espec_isco()

double Lorene::Compobj_QI::espec_isco ( int lmin ) const

virtualinherited

Energy of a particle at the ISCO.

Definition at line 320 of file compobj_QI_global.C.

References p_espec_isco, and r_isco().

◆ extrinsic_curvature()

void Lorene::Compobj_QI::extrinsic_curvature ( )

virtualinherited

Computes the extrinsic curvature and ak_car from nphi , nn and b_car .

Reimplemented from Lorene::Compobj.

Reimplemented in Lorene::AltBH_QI.

Definition at line 339 of file compobj_QI.C.

References ak_car, b_car, del_deriv(), Lorene::Compobj::extrinsic_curvature(), Lorene::Compobj::kk, Lorene::Compobj::mp, Lorene::Scalar::mult_rsint(), Lorene::Scalar::mult_sint(), Lorene::Compobj::nn, and nphi.

◆ f_isco()

double Lorene::Compobj_QI::f_isco ( int lmin ) const

virtualinherited

Orbital frequency at the innermost stable circular orbit (ISCO).

Definition at line 286 of file compobj_QI_global.C.

References p_f_isco, and r_isco().

◆ get_a_car()

const Scalar & Lorene::Compobj_QI::get_a_car ( ) const

inlineinherited

Returns the square of the metric factor A.

Definition at line 378 of file compobj.h.

References a_car.

◆ get_ak_car()

const Scalar & Lorene::Compobj_QI::get_ak_car ( ) const

inlineinherited

Returns the scalar $A^2 K_{ij} K^{ij}$ .

For axisymmetric stars, this quantity is related to the derivatives of $N^\varphi$ by

$\‍[ A^2 K_{ij} K^{ij} = {B^2 \over 2 N^2} \, r^2\sin^2\theta \, \left[ \left( {\partial N^\varphi \over \partial r} \right) ^2 + {1\over r^2} \left( {\partial N^\varphi \over \partial \theta} \right) ^2 \right] \ . \‍]$

In particular it is related to the quantities $k_1$ and $k_2$ introduced by Eqs. (3.7) and (3.8) of Bonazzola et al. Astron. Astrophys. 278 , 421 (1993) by

$\‍[ A^2 K_{ij} K^{ij} = 2 A^2 (k_1^2 + k_2^2) \ . \‍]$

Definition at line 404 of file compobj.h.

References ak_car.

◆ get_b_car()

const Scalar & Lorene::Compobj_QI::get_b_car ( ) const

inlineinherited

Returns the square of the metric factor B.

Definition at line 381 of file compobj.h.

References b_car.

◆ get_bbb()

const Scalar & Lorene::Compobj_QI::get_bbb ( ) const

inlineinherited

Returns the metric factor B.

Definition at line 375 of file compobj.h.

References bbb.

◆ get_beta()

const Vector & Lorene::Compobj::get_beta ( ) const

inlineinherited

Returns the shift vector $\beta^i$ .

Definition at line 213 of file compobj.h.

References beta.

◆ get_ener_euler()

const Scalar & Lorene::Compobj::get_ener_euler ( ) const

inlineinherited

Returns the total energy density E in the Eulerian frame.

Definition at line 219 of file compobj.h.

References ener_euler.

◆ get_gamma()

const Metric & Lorene::Compobj::get_gamma ( ) const

inlineinherited

Returns the 3-metric $\gamma_{ij}$ .

Definition at line 216 of file compobj.h.

References gamma.

◆ get_kk()

const Sym_tensor & Lorene::Compobj::get_kk ( ) const

inlineinherited

Returns the extrinsic curvature tensor $K_{ij}$ .

Definition at line 228 of file compobj.h.

References kk.

◆ get_mom_euler()

const Vector & Lorene::Compobj::get_mom_euler ( ) const

inlineinherited

Returns the total 3-momentum density $P^i$ in the Eulerian frame.

Definition at line 222 of file compobj.h.

References mom_euler.

◆ get_mp()

const Map & Lorene::Compobj::get_mp ( ) const

inlineinherited

Returns the mapping.

Definition at line 207 of file compobj.h.

References Lorene::Map(), and mp.

◆ get_nn()

const Scalar & Lorene::Compobj::get_nn ( ) const

inlineinherited

Returns the lapse function N .

Definition at line 210 of file compobj.h.

References nn.

◆ get_nphi()

const Scalar & Lorene::Compobj_QI::get_nphi ( ) const

inlineinherited

Returns the metric coefficient $N^\varphi$ .

Definition at line 384 of file compobj.h.

References nphi.

◆ get_stress_euler()

const Sym_tensor & Lorene::Compobj::get_stress_euler ( ) const

inlineinherited

Returns the stress tensor $S_{ij}$ with respect to the Eulerian observer.

Definition at line 225 of file compobj.h.

References stress_euler.

◆ gyoto_data()

void Lorene::Compobj_QI::gyoto_data ( const char * file_name ) const

inherited

Save in a file for GYOTO.

Definition at line 232 of file compobj_QI.C.

References Lorene::Compobj::beta, Lorene::fwrite_be(), Lorene::Compobj::gamma, Lorene::Compobj::kk, Lorene::Compobj::mp, Lorene::Compobj::nn, r_isco(), and r_mb().

◆ lspec_isco()

double Lorene::Compobj_QI::lspec_isco ( int lmin ) const

virtualinherited

Angular momentum of a particle at the ISCO.

Definition at line 303 of file compobj_QI_global.C.

References p_lspec_isco, and r_isco().

◆ operator=()

void Lorene::Kerr_QI::operator= ( const Kerr_QI & other )

Assignment to another Kerr_QI.

Definition at line 178 of file kerr_QI.C.

References aa, del_deriv(), Kerr_QI(), mm, and Lorene::Compobj_QI::operator=().

◆ operator>>()

ostream & Lorene::Kerr_QI::operator>> ( ostream & ost ) const

protectedvirtual

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

Reimplemented from Lorene::Compobj_QI.

Definition at line 203 of file kerr_QI.C.

References aa, mm, and Lorene::Compobj_QI::operator>>().

◆ r_isco()

double Lorene::Compobj_QI::r_isco	(	int	lmin,
		ostream *	ost = 0x0 ) const

virtualinherited

Coordinate r of the innermost stable circular orbit (ISCO).

Parameters

lmin	index of the innermost domain in which the ISCO is searched: the ISCO is searched inwards from the last but one domain to the domain of index lmin.
ost	output stream to give details of the computation; if set to 0x0 [default value], no details will be given.

Definition at line 108 of file compobj_QI_global.C.

References Lorene::Param::add_int(), Lorene::Param::add_scalar(), Lorene::Tensor::annule_domain(), bbb, Lorene::Scalar::dsdr(), Lorene::Scalar::get_spectral_va(), Lorene::Compobj::mp, Lorene::Compobj::nn, nphi, p_espec_isco, p_f_isco, p_lspec_isco, p_r_isco, Lorene::r, Lorene::sqrt(), Lorene::Scalar::std_spectral_base(), Lorene::Valeur::val_point(), and Lorene::zerosec().

◆ r_mb()

double Lorene::Compobj_QI::r_mb	(	int	lmin,
		ostream *	ost = 0x0 ) const

virtualinherited

Coordinate r of the marginally bound circular orbit (R_mb).

Definition at line 341 of file compobj_QI_global.C.

References Lorene::Param::add_int(), Lorene::Param::add_scalar(), Lorene::Tensor::annule_domain(), bbb, Lorene::Scalar::get_spectral_va(), Lorene::Compobj::mp, Lorene::Compobj::nn, nphi, p_r_mb, Lorene::r, r_isco(), Lorene::sqrt(), Lorene::Scalar::std_spectral_base(), Lorene::Valeur::val_point(), and Lorene::zerosec().

◆ sauve()

void Lorene::Kerr_QI::sauve ( FILE * fich ) const

virtual

Save in a file.

Reimplemented from Lorene::Compobj_QI.

Definition at line 195 of file kerr_QI.C.

◆ set_der_0x0()

void Lorene::Kerr_QI::set_der_0x0 ( ) const

protectedvirtual

Sets to 0x0 all the pointers on derived quantities.

Definition at line 168 of file kerr_QI.C.

◆ set_mp()

Map & Lorene::Compobj::set_mp ( )

inlineinherited

Read/write of the mapping.

Definition at line 200 of file compobj.h.

References Lorene::Map(), and mp.

◆ update_metric()

void Lorene::Compobj_QI::update_metric ( )

virtualinherited

Updates the 3-metric $\gamma_{ij}$ from A and B and the shift vector $\beta^i$ from $N^\phi$ .

Reimplemented in Lorene::Star_QI.

Definition at line 302 of file compobj_QI.C.

References a_car, b_car, Lorene::Compobj::beta, del_deriv(), extrinsic_curvature(), Lorene::Compobj::gamma, Lorene::Compobj::mp, Lorene::Scalar::mult_rsint(), nphi, and Lorene::Tensor::set().

Member Data Documentation

◆ a_car

Scalar Lorene::Compobj_QI::a_car

protectedinherited

Square of the metric factor A.

Definition at line 287 of file compobj.h.

◆ aa

double Lorene::Kerr_QI::aa

protected

angular momentum parameter $a$

Definition at line 828 of file compobj.h.

◆ ak_car

Scalar Lorene::Compobj_QI::ak_car

protectedinherited

Scalar $A^2 K_{ij} K^{ij}$ .

For axisymmetric stars, this quantity is related to the derivatives of $N^\varphi$ by

$\‍[ A^2 K_{ij} K^{ij} = {B^2 \over 2 N^2} \, r^2\sin^2\theta \, \left[ \left( {\partial N^\varphi \over \partial r} \right) ^2 + {1\over r^2} \left( {\partial N^\varphi \over \partial \theta} \right) ^2 \right] \ . \‍]$

In particular it is related to the quantities $k_1$ and $k_2$ introduced by Eqs.~(3.7) and (3.8) of Bonazzola et al. Astron. Astrophys. 278 , 421 (1993) by

$\‍[ A^2 K_{ij} K^{ij} = 2 A^2 (k_1^2 + k_2^2) \ . \‍]$

Definition at line 315 of file compobj.h.

◆ b_car

Scalar Lorene::Compobj_QI::b_car

protectedinherited

Square of the metric factor B.

Definition at line 293 of file compobj.h.

◆ bbb

Scalar Lorene::Compobj_QI::bbb

protectedinherited

Metric factor B.

Definition at line 290 of file compobj.h.

◆ beta

Vector Lorene::Compobj::beta

protectedinherited

Shift vector $\beta^i$ .

Definition at line 138 of file compobj.h.

◆ ener_euler

Scalar Lorene::Compobj::ener_euler

protectedinherited

Total energy density E in the Eulerian frame.

Definition at line 144 of file compobj.h.

◆ gamma

Metric Lorene::Compobj::gamma

protectedinherited

3-metric $\gamma_{ij}$

Definition at line 141 of file compobj.h.

◆ kk

Sym_tensor Lorene::Compobj::kk

protectedinherited

Extrinsic curvature tensor $K_{ij}$ .

Definition at line 153 of file compobj.h.

◆ mm

double Lorene::Kerr_QI::mm

protected

mass parameter $M$

Definition at line 824 of file compobj.h.

◆ mom_euler

Vector Lorene::Compobj::mom_euler

protectedinherited

Total 3-momentum density $P^i$ in the Eulerian frame.

Definition at line 147 of file compobj.h.

◆ mp

Map& Lorene::Compobj::mp

protectedinherited

Mapping describing the coordinate system (r,theta,phi).

Definition at line 132 of file compobj.h.

◆ nn

Scalar Lorene::Compobj::nn

protectedinherited

Lapse function N .

Definition at line 135 of file compobj.h.

◆ nphi

Scalar Lorene::Compobj_QI::nphi

protectedinherited

Metric coefficient $N^\varphi$ .

Definition at line 296 of file compobj.h.

◆ p_adm_mass

double* Lorene::Compobj::p_adm_mass

mutableprotectedinherited

ADM mass.

Definition at line 158 of file compobj.h.

◆ p_angu_mom

double* Lorene::Compobj_QI::p_angu_mom

mutableprotectedinherited

Angular momentum.

Definition at line 321 of file compobj.h.

◆ p_espec_isco

double* Lorene::Compobj_QI::p_espec_isco

mutableprotectedinherited

Specific energy of a particle at the ISCO.

Definition at line 325 of file compobj.h.

◆ p_f_isco

double* Lorene::Compobj_QI::p_f_isco

mutableprotectedinherited

Orbital frequency of the ISCO.

Definition at line 323 of file compobj.h.

◆ p_lspec_isco

double* Lorene::Compobj_QI::p_lspec_isco

mutableprotectedinherited

Specific angular momentum of a particle at the ISCO.

Definition at line 327 of file compobj.h.

◆ p_r_isco

double* Lorene::Compobj_QI::p_r_isco

mutableprotectedinherited

Coordinate r of the ISCO.

Definition at line 322 of file compobj.h.

◆ p_r_mb

double* Lorene::Compobj_QI::p_r_mb

mutableprotectedinherited

Coordinate r of the marginally bound orbit.

Definition at line 328 of file compobj.h.

◆ stress_euler

Sym_tensor Lorene::Compobj::stress_euler

protectedinherited

Stress tensor $S_{ij}$ with respect to the Eulerian observer.

Definition at line 150 of file compobj.h.

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

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ Kerr_QI() [1/3]

◆ Kerr_QI() [2/3]

◆ Kerr_QI() [3/3]

◆ ~Kerr_QI()

Member Function Documentation

◆ adm_mass()

◆ angu_mom()

◆ del_deriv()

◆ espec_isco()

◆ extrinsic_curvature()

◆ f_isco()

◆ get_a_car()

◆ get_ak_car()

◆ get_b_car()

◆ get_bbb()

◆ get_beta()

◆ get_ener_euler()

◆ get_gamma()

◆ get_kk()

◆ get_mom_euler()

◆ get_mp()

◆ get_nn()

◆ get_nphi()

◆ get_stress_euler()

◆ gyoto_data()

◆ lspec_isco()

◆ operator=()

◆ operator>>()

◆ r_isco()

◆ r_mb()

◆ sauve()

◆ set_der_0x0()

◆ set_mp()

◆ update_metric()

Member Data Documentation

◆ a_car

◆ aa

◆ ak_car

◆ b_car

◆ bbb

◆ beta

◆ ener_euler

◆ gamma

◆ kk

◆ mm

◆ mom_euler

◆ mp

◆ nn

◆ nphi

◆ p_adm_mass

◆ p_angu_mom

◆ p_espec_isco

◆ p_f_isco

◆ p_lspec_isco

◆ p_r_isco

◆ p_r_mb

◆ stress_euler