Spheroid Class Reference
[Stars and black holes]

Spheroidal 2-surfaces embedded in a time-slice of the 3+1 formalism. More...

#include <spheroid.h>

Inheritance diagram for Spheroid:

Public Member Functions
	Spheroid (const Map_af &map, double radius)
	The delta tensorial fields linked to Christoffel symbols.
	Spheroid (const Scalar &h_in, const Metric &gamij, const Sym_tensor &Kij)
	Constructor of a spheroid embedded in a 3-slice (`Time_slice` ) of 3+1 formalism.
	Spheroid (const Spheroid &)
	Copy constructor.
	Spheroid (FILE *)
	Constructor from a file (see `sauve(FILE*)` ).
virtual	~Spheroid ()
	Destructor.
void	operator= (const Spheroid &)
	Assignment to another Spheroid.
void	set_ephi (const Scalar &)
	Assigns the conformal Killing vector field to phi.
const Scalar &	get_hsurf () const
	Returns the field `h_surf`.
const Metric &	get_qab () const
	Returns the metric $q_{ab}$ .
const Scalar &	get_ricci () const
	Returns the 2-ricci scalar $R = q^{ab}R_{ab}$ .
const Sym_tensor &	get_hh () const
	Returns the symmetric tensor $H_{ab}$ .
const Sym_tensor &	get_qq () const
	returns the 3-d degenerate 2-metric $Q_{ab}$
const Tensor &	get_proj () const
	returns the 3-d projector on 2-surface
const Tensor &	get_jac2d () const
	returns the 2-d jacobian of coordinate transformation
const Scalar &	get_trk () const
	Returns the trace `K` on the 2-surface.
const Vector &	get_ll () const
	Returns the vector .
const Vector &	get_ss () const
	Returns the vector .
const Vector &	get_ephi () const
	Returns the conformal Killing symmetry vector on the 2-surface.
const Sym_tensor &	get_jj () const
	Returns the symmetric tensor $J_{ab}$ .
const Scalar &	get_fff () const
	Returns the normalization scalar `F`.
const Scalar &	get_ggg () const
	Returns the normalization scalar `G`.
bool	get_issphere () const
	Returns the flag saying whether or not the horizon is geometrically round.
Scalar &	set_hsurf ()
	Sets the field `h_surf`.
Metric &	set_qab ()
	Sets the modified metric (non degenerated) $q_{ab}$ .
Scalar &	set_ricci ()
	Sets the 2-Ricci scalar $R = q^{ab}R_{ab}$ .
Sym_tensor &	set_qq ()
	Sets the degenerated metric $Q_{ab}$ .
Tensor &	set_proj ()
	Sets the projector .
Sym_tensor &	set_hh ()
	Sets the symmetric tensor $H_{ab}$ .
Scalar &	set_trk ()
	Sets the trace `K` on the 2-surface.
Vector &	set_ll ()
	Sets the vector .
Vector &	set_ss ()
	Sets the vector .
Sym_tensor &	set_jj ()
	Sets the symmetric tensor $J_{ab}$ .
Scalar &	set_fff ()
	Sets the normalization factor `F`.
Scalar &	set_ggg ()
	Sets the normalization factor `G`.
bool	set_issphere ()
	Sets the boolean linked to geometrical shape of the horizon.
void	update_from_tslice (const Metric &gamij, const Sym_tensor &Kij)
	Updates from the 3-slice data.
const Scalar &	sqrt_q () const
	Computes the normal vector field to the 2-surface.
double	area () const
	Computes the area of the 2-surface.
double	angu_mom () const
	Computes the angular momentum with respect to a divergence-free vector field tangent to the 2-surface.
double	mass () const
	Computes the mass as defined from the calculus of angular momentum, done with respect to a divergence free tangent vector field .
double	multipole_mass (const int order) const
	Computes the mass multipole of a given order for the spheroid, assumed to be spherical.
double	multipole_angu (const int order) const
	Computes the angular multipole of a given order for the spheroid, assumed to be spherical.
double	epsilon_A_minus_one () const
	Computation of the refined Penrose parameter for axisymmetric spacetimes, and its difference wrt one.
double	epsilon_P_minus_one () const
	Computation of the classical Penrose parameter, and its difference wrt one.
const Scalar &	theta_plus () const
	Computes the outgoing null expansion $\theta_+$ .
const Scalar &	theta_minus () const
	Computes the ingoing null expansion $\theta_-$ .
const Sym_tensor &	shear () const
	Computes the shear of the 2-surface $\sigma_{ab}$ .
Tensor	derive_cov2dflat (const Tensor &uu) const
	Computes the round covariant derivative on the spheroid.
const Tensor &	delta () const
	Computes the delta coefficients for covariant derivative.
Tensor	derive_cov2d (const Tensor &uu) const
	Computes the total covariant derivative on the spheroid.
virtual void	sauve (FILE *) const
	Save in a file.
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.
Protected Attributes
Scalar	h_surf
	The location of the 2-surface as r = `h_surf`.
Tensor	jac2d
	The jacobian of the adaptation of coordinates (contravariant/covariant representation).
Tensor	proj
	The 3-d projector on the 2-surface (contravariant-covariant form).
Sym_tensor	qq
	The 3-d covariant degenerated 2-metric on the surface.
Vector	ss
	The adapted normal vector field to spheroid in the 3-slice.
Vector	ephi
	The conformal Killing vector field on the 2-surface (set to by default to the axial vector associated with coordinate $\varphi$ ).
Metric	qab
Scalar	ricci
	Induced metric on the 2-surface $q_{ab}$ .
Sym_tensor	hh
	The ricci scalar on the 2-surface.
Scalar	trk
	Trace K of the extrinsic curvature of the 3-slice.
Vector	ll
	Normal-tangent component of the extrinsic curvature of the 3-slice.
Sym_tensor	jj
	Tangent components of the extrinsic curvature of the 3-slice.
Scalar	fff
	Normalization function for the outgoing null vector l.
Scalar	ggg
	Normalization function for the ingoing null vector k.
Scalar	zeta
bool	issphere
	Flag to know whether the horizon is geometrically round or distorted.
Scalar *	p_sqrt_q
	Surface element $\sqrt{\det q_{ab}}$ .
double *	p_area
	The area of the 2-surface.
double *	p_angu_mom
	The angular momentum.
double *	p_mass
	Mass defined from angular momentum.
double *	p_multipole_mass
	Mass multipole for the spheroid.
double *	p_multipole_angu
	Angular momentum multipole for the spheroid.
double *	p_epsilon_A_minus_one
double *	p_epsilon_P_minus_one
	Refined Penrose parameter, difference wrt one.
Scalar *	p_theta_plus
	Classical Penrose parameter, difference wrt one.
Scalar *	p_theta_minus
	Null ingoing expansion.
Sym_tensor *	p_shear
	The shear tensor.
Tensor *	p_delta
Friends
ostream &	operator<< (ostream &, const Spheroid &)
	Display.

Detailed Description

Spheroidal 2-surfaces embedded in a time-slice of the 3+1 formalism.

()

Definition at line 80 of file spheroid.h.

Constructor & Destructor Documentation

Spheroid::Spheroid	(	const Map_af &	map,
		double	radius
	)

The delta tensorial fields linked to Christoffel symbols.

Standard constructor. The input mapping must be defined on a mono-domain angular grid (see Mg3d::get_angu_mono_domain() for details).

Definition at line 90 of file spheroid.C.

References ephi, fff, Map::get_mg(), Mg3d::get_nr(), Mg3d::get_nzone(), Mg3d::get_type_r(), ggg, h_surf, hh, jac2d, jj, ll, proj, Tensor::set(), set_der_0x0(), Scalar::set_etat_zero(), Tensor::set_etat_zero(), Tensor::set_index_type(), ss, and trk.

Spheroid::Spheroid	(	const Scalar &	h_in,
		const Metric &	gamij,
		const Sym_tensor &	Kij
	)

Constructor of a spheroid embedded in a 3-slice (Time_slice ) of 3+1 formalism.

This is done from the Time_slice data.

Parameters:

	h_in	: the location of the surface r = h_in (WARNING:must be defined on a mono-domain angular grid)
	gamij	: the 3-metric on the 3-slice
	Kij	: the extrinsic curvature of the 3-slice (covariant representation)

Definition at line 146 of file spheroid.C.

References Tensor::allocate_all(), Scalar::allocate_all(), Tensor::annule_domain(), area(), Valeur::c_cf, Valeur::coef(), Metric::con(), contract(), Metric::cov(), Scalar::dec_dzpuis(), del_deriv(), Scalar::derive_con(), Scalar::derive_cov(), Tensor::down(), ephi, fff, Mg3d::get_angu_1dom(), Map::get_bvect_spher(), Map::get_mg(), Metric::get_mp(), Tensor::get_mp(), Mg3d::get_np(), Mg3d::get_nr(), Mg3d::get_nt(), Mg3d::get_nzone(), Scalar::get_spectral_base(), Scalar::get_spectral_va(), ggg, Base_val::give_quant_numbers(), h_surf, hh, jac2d, jj, ll, Scalar::mult_cost(), proj, qq, Map::r, ricci, Metric::ricci(), Vector::set(), Tensor::set(), Itbl::set(), set_der_0x0(), Scalar::set_grid_point(), Tensor::set_index_type(), Scalar::set_spectral_va(), sqrt(), sqrt_q(), Scalar::srdsdt(), Scalar::srstdsdp(), ss, Vector::std_spectral_base(), Tensor::std_spectral_base(), Scalar::std_spectral_base(), Tensor::trace(), trk, Tensor::up(), Scalar::val_grid_point(), Map_radial::val_lx_jk(), Valeur::val_point_jk(), and Valeur::ylm().

Spheroid::Spheroid ( const Spheroid & sph_in )

Copy constructor.

Definition at line 587 of file spheroid.C.

References set_der_0x0().

Spheroid::Spheroid ( FILE * )

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

Spheroid::~Spheroid ( ) [virtual]

Destructor.

Definition at line 638 of file spheroid.C.

References del_deriv().

Member Function Documentation

double Spheroid::angu_mom ( ) const

Computes the angular momentum with respect to a divergence-free vector field tangent to the 2-surface.

This is defined as

${\cal J} = \int h^2 L_i \phi^i \sqrt{\det q_{ab}} \sin \theta {\rm d}\theta {\rm d}\varphi$

Parameters:

phi

: the divergence-free vector field $\phi$

Definition at line 717 of file spheroid.C.

References contract(), Map::get_bvect_spher(), get_ephi(), Tensor::get_mp(), h_surf, Map_af::integrale_surface(), jac2d, ll, p_angu_mom, and sqrt_q().

double Spheroid::area ( ) const

Computes the area of the 2-surface.

This is defined as

${\cal A} = \int h^2 \sqrt{\det q_{ab}} \sin \theta {\rm d}\theta {\rm d}\varphi$

Definition at line 704 of file spheroid.C.

References Tensor::get_mp(), h_surf, Map_af::integrale_surface(), p_area, and sqrt_q().

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

Deletes all the derived quantities.

Definition at line 646 of file spheroid.C.

References p_angu_mom, p_area, p_epsilon_P_minus_one, p_mass, p_multipole_angu, p_multipole_mass, p_shear, p_sqrt_q, p_theta_minus, p_theta_plus, and set_der_0x0().

const Tensor & Spheroid::delta ( ) const

Computes the delta coefficients for covariant derivative.

Definition at line 1199 of file spheroid.C.

References Metric::con(), Metric::cov(), derive_cov2dflat(), Map::get_bvect_spher(), Metric::get_mp(), Tensor::set(), and Tensor::set_index_type().

Tensor Spheroid::derive_cov2d ( const Tensor & uu ) const

Computes the total covariant derivative on the spheroid.

Definition at line 1236 of file spheroid.C.

References contract(), delta(), derive_cov2dflat(), Tensor::get_index_type(), and Tensor::get_valence().

Tensor Spheroid::derive_cov2dflat ( const Tensor & uu ) const

Computes the round covariant derivative on the spheroid.

Definition at line 950 of file spheroid.C.

References Map::get_bvect_spher(), Tensor::get_index_type(), Tensor::get_mp(), Tensor::get_n_comp(), Tensor::get_triad(), Tensor::get_valence(), Tensor::indices(), Tensor::set(), Itbl::set(), Tensor_sym::sym_index1(), and Tensor_sym::sym_index2().

double Spheroid::epsilon_A_minus_one ( ) const

Computation of the refined Penrose parameter for axisymmetric spacetimes, and its difference wrt one.

Definition at line 860 of file spheroid.C.

References angu_mom(), area(), mass(), pow(), and sqrt().

double Spheroid::epsilon_P_minus_one ( ) const

Computation of the classical Penrose parameter, and its difference wrt one.

To use in replacement of epsilon_A_minus_one when the computed spacetime is not axisymmetric.

Definition at line 872 of file spheroid.C.

References angu_mom(), area(), mass(), p_epsilon_P_minus_one, and pow().

const Vector& Spheroid::get_ephi ( ) const [inline]

Returns the conformal Killing symmetry vector on the 2-surface.

Definition at line 249 of file spheroid.h.

References ephi.

const Scalar& Spheroid::get_fff ( ) const [inline]

Returns the normalization scalar F.

Definition at line 255 of file spheroid.h.

References fff.

const Scalar& Spheroid::get_ggg ( ) const [inline]

Returns the normalization scalar G.

Definition at line 258 of file spheroid.h.

References ggg.

const Sym_tensor& Spheroid::get_hh ( ) const [inline]

Returns the symmetric tensor $H_{ab}$ .

Definition at line 228 of file spheroid.h.

References hh.

const Scalar& Spheroid::get_hsurf ( ) const [inline]

Returns the field h_surf.

Definition at line 219 of file spheroid.h.

References h_surf.

bool Spheroid::get_issphere ( ) const [inline]

Returns the flag saying whether or not the horizon is geometrically round.

Definition at line 261 of file spheroid.h.

References issphere.

const Tensor& Spheroid::get_jac2d ( ) const [inline]

returns the 2-d jacobian of coordinate transformation $ J $

Definition at line 237 of file spheroid.h.

References jac2d.

const Sym_tensor& Spheroid::get_jj ( ) const [inline]

Returns the symmetric tensor $J_{ab}$ .

Definition at line 252 of file spheroid.h.

References jj.

const Vector& Spheroid::get_ll ( ) const [inline]

Returns the vector $ L_a $ .

Definition at line 243 of file spheroid.h.

References ll.

const Tensor& Spheroid::get_proj ( ) const [inline]

returns the 3-d projector on 2-surface $ Pi $

Definition at line 234 of file spheroid.h.

References proj.

const Metric& Spheroid::get_qab ( ) const [inline]

Returns the metric $q_{ab}$ .

Definition at line 222 of file spheroid.h.

const Sym_tensor& Spheroid::get_qq ( ) const [inline]

returns the 3-d degenerate 2-metric $Q_{ab}$

Definition at line 231 of file spheroid.h.

References qq.

const Scalar& Spheroid::get_ricci ( ) const [inline]

Returns the 2-ricci scalar $R = q^{ab}R_{ab}$ .

Definition at line 225 of file spheroid.h.

References ricci.

const Vector& Spheroid::get_ss ( ) const [inline]

Returns the vector $ S_a $ .

Definition at line 246 of file spheroid.h.

References ss.

const Scalar& Spheroid::get_trk ( ) const [inline]

Returns the trace K on the 2-surface.

Definition at line 240 of file spheroid.h.

References trk.

double Spheroid::mass ( ) const

Computes the mass as defined from the calculus of angular momentum, done with respect to a divergence free tangent vector field $ phi $ .

Spheroid has to be a real sphere (flag issphere true), of constant radius

$R_{s}$

. defined as

$M = \frac{1}{2 R_{s}} \sqrt{R_{s}^{4} + 4{\cal J}^{2}}$

Definition at line 732 of file spheroid.C.

References angu_mom(), area(), p_mass, and sqrt().

double Spheroid::multipole_angu ( const int order ) const

Computes the angular multipole of a given order for the spheroid, assumed to be spherical.

$ phi $ is a divergence free tangent vector field. WARNING:order has to be strictly higher than zero (no topological defects here...), and an odd number for technical reasons.

Definition at line 797 of file spheroid.C.

References area(), contract(), Map::get_bvect_spher(), get_ephi(), Tensor::get_mp(), h_surf, Map_af::integrale_surface(), jac2d, ll, p_multipole_angu, Scalar::set_spectral_va(), sqrt(), sqrt_q(), Scalar::std_spectral_base(), and Valeur::ylm().

double Spheroid::multipole_mass ( const int order ) const

Computes the mass multipole of a given order for the spheroid, assumed to be spherical.

WARNING: For technical reasons, only even orders are supported by the code.

Definition at line 744 of file spheroid.C.

References area(), Tensor::get_mp(), get_ricci(), h_surf, Map_af::integrale_surface(), mass(), p_multipole_mass, Scalar::set_spectral_va(), sqrt(), sqrt_q(), Scalar::std_spectral_base(), and Valeur::ylm().

void Spheroid::operator= ( const Spheroid & sph_in )

Assignment to another Spheroid.

Reimplemented in App_hor.

Definition at line 610 of file spheroid.C.

References del_deriv(), ephi, fff, ggg, h_surf, hh, issphere, jac2d, jj, ll, proj, qq, ricci, ss, and trk.

void Spheroid::sauve ( FILE * ) const [virtual]

Save in a file.

Definition at line 1182 of file spheroid.C.

void Spheroid::set_der_0x0 ( ) const [protected]

Sets to 0x0 all the pointers on derived quantities.

Definition at line 662 of file spheroid.C.

References p_angu_mom, p_area, p_epsilon_P_minus_one, p_mass, p_multipole_angu, p_multipole_mass, p_shear, p_sqrt_q, p_theta_minus, and p_theta_plus.

void Spheroid::set_ephi ( const Scalar & )

Assigns the conformal Killing vector field to phi.

Scalar& Spheroid::set_fff ( ) [inline]

Sets the normalization factor F.

Definition at line 294 of file spheroid.h.

References del_deriv(), and fff.

Scalar& Spheroid::set_ggg ( ) [inline]

Sets the normalization factor G.

Definition at line 297 of file spheroid.h.

References del_deriv(), and ggg.

Sym_tensor& Spheroid::set_hh ( ) [inline]

Sets the symmetric tensor $H_{ab}$ .

Definition at line 279 of file spheroid.h.

References del_deriv(), and hh.

Scalar& Spheroid::set_hsurf ( ) [inline]

Sets the field h_surf.

Definition at line 264 of file spheroid.h.

References del_deriv(), and h_surf.

bool Spheroid::set_issphere ( ) [inline]

Sets the boolean linked to geometrical shape of the horizon.

Definition at line 300 of file spheroid.h.

References del_deriv(), and issphere.

Sym_tensor& Spheroid::set_jj ( ) [inline]

Sets the symmetric tensor $J_{ab}$ .

Definition at line 291 of file spheroid.h.

References del_deriv(), and jj.

Vector& Spheroid::set_ll ( ) [inline]

Sets the vector $ L_a $ .

Definition at line 285 of file spheroid.h.

References del_deriv(), and ll.

Tensor& Spheroid::set_proj ( ) [inline]

Sets the projector $ Pi $ .

Definition at line 276 of file spheroid.h.

References del_deriv(), and proj.

Metric& Spheroid::set_qab ( ) [inline]

Sets the modified metric (non degenerated) $q_{ab}$ .

Definition at line 267 of file spheroid.h.

References del_deriv().

Sym_tensor& Spheroid::set_qq ( ) [inline]

Sets the degenerated metric $Q_{ab}$ .

Definition at line 273 of file spheroid.h.

References del_deriv(), and qq.

Scalar& Spheroid::set_ricci ( ) [inline]

Sets the 2-Ricci scalar $R = q^{ab}R_{ab}$ .

Definition at line 270 of file spheroid.h.

References del_deriv(), and ricci.

Vector& Spheroid::set_ss ( ) [inline]

Sets the vector $ s_a $ .

Definition at line 288 of file spheroid.h.

References del_deriv(), and ss.

Scalar& Spheroid::set_trk ( ) [inline]

Sets the trace K on the 2-surface.

Definition at line 282 of file spheroid.h.

References del_deriv(), and trk.

const Sym_tensor & Spheroid::shear ( ) const

Computes the shear of the 2-surface $\sigma_{ab}$ .

Definition at line 921 of file spheroid.C.

References Metric::cov(), fff, hh, jj, p_shear, Tensor::std_spectral_base(), and Tensor::trace().

const Scalar & Spheroid::sqrt_q ( ) const

Computes the normal vector field to the 2-surface.

Computes the square root of the determinant of $q_{ab}$ .

Definition at line 689 of file spheroid.C.

References get_qq(), p_sqrt_q, sqrt(), and Scalar::std_spectral_base().

const Scalar & Spheroid::theta_minus ( ) const

Computes the ingoing null expansion $\theta_-$ .

Definition at line 905 of file spheroid.C.

References ggg, hh, jj, p_theta_minus, Scalar::std_spectral_base(), and Tensor::trace().

const Scalar & Spheroid::theta_plus ( ) const

Computes the outgoing null expansion $\theta_+$ .

Definition at line 885 of file spheroid.C.

References fff, hh, jj, p_theta_plus, Scalar::set_spectral_va(), Scalar::std_spectral_base(), Tensor::trace(), and Valeur::ylm().

void Spheroid::update_from_tslice	(	const Metric &	gamij,
		const Sym_tensor &	Kij
	)

Updates from the 3-slice data.

Friends And Related Function Documentation

ostream& operator<<	(	ostream &	,
		const Spheroid &
	)			`[friend]`

Display.

Member Data Documentation

Vector Spheroid::ephi [protected]

The conformal Killing vector field on the 2-surface (set to by default to the axial vector associated with coordinate $\varphi$ ).

Definition at line 109 of file spheroid.h.

Scalar Spheroid::fff [protected]

Normalization function for the outgoing null vector l.

Definition at line 134 of file spheroid.h.

Scalar Spheroid::ggg [protected]

Normalization function for the ingoing null vector k.

Definition at line 139 of file spheroid.h.

Scalar Spheroid::h_surf [protected]

The location of the 2-surface as r = h_surf.

Definition at line 87 of file spheroid.h.

Sym_tensor Spheroid::hh [protected]

The ricci scalar on the 2-surface.

Extrinsic curvature of the 2-surface in the 3-slice. $H_{ab}$ (covariant representation)

Definition at line 118 of file spheroid.h.

bool Spheroid::issphere [protected]

Flag to know whether the horizon is geometrically round or distorted.

Definition at line 147 of file spheroid.h.

Tensor Spheroid::jac2d [protected]

The jacobian of the adaptation of coordinates (contravariant/covariant representation).

Definition at line 92 of file spheroid.h.

Sym_tensor Spheroid::jj [protected]

Tangent components of the extrinsic curvature of the 3-slice.

$J_{ab}$ (covariant representation)

Definition at line 130 of file spheroid.h.

Vector Spheroid::ll [protected]

Normal-tangent component of the extrinsic curvature of the 3-slice.

$ L_a $ (covariant representation)

Definition at line 125 of file spheroid.h.

double* Spheroid::p_angu_mom [mutable, protected]

The angular momentum.

Definition at line 155 of file spheroid.h.

double* Spheroid::p_area [mutable, protected]

The area of the 2-surface.

Definition at line 154 of file spheroid.h.

double* Spheroid::p_epsilon_P_minus_one [mutable, protected]

Refined Penrose parameter, difference wrt one.

Definition at line 160 of file spheroid.h.

double* Spheroid::p_mass [mutable, protected]

Mass defined from angular momentum.

Definition at line 156 of file spheroid.h.

double* Spheroid::p_multipole_angu [mutable, protected]

Angular momentum multipole for the spheroid.

Definition at line 158 of file spheroid.h.

double* Spheroid::p_multipole_mass [mutable, protected]

Mass multipole for the spheroid.

Definition at line 157 of file spheroid.h.

Sym_tensor* Spheroid::p_shear [mutable, protected]

The shear tensor.

Definition at line 163 of file spheroid.h.

Scalar* Spheroid::p_sqrt_q [mutable, protected]

Surface element $\sqrt{\det q_{ab}}$ .

Definition at line 153 of file spheroid.h.

Scalar* Spheroid::p_theta_minus [mutable, protected]

Null ingoing expansion.

Definition at line 162 of file spheroid.h.

Scalar* Spheroid::p_theta_plus [mutable, protected]

Classical Penrose parameter, difference wrt one.

Null outgoing expansion

Definition at line 161 of file spheroid.h.

Tensor Spheroid::proj [protected]

The 3-d projector on the 2-surface (contravariant-covariant form).

Definition at line 96 of file spheroid.h.

Sym_tensor Spheroid::qq [protected]

The 3-d covariant degenerated 2-metric on the surface.

Definition at line 100 of file spheroid.h.

Scalar Spheroid::ricci [protected]

Induced metric on the 2-surface $q_{ab}$ .

Definition at line 113 of file spheroid.h.

Vector Spheroid::ss [protected]

The adapted normal vector field to spheroid in the 3-slice.

Definition at line 104 of file spheroid.h.

Scalar Spheroid::trk [protected]

Trace K of the extrinsic curvature of the 3-slice.

Definition at line 120 of file spheroid.h.

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

Spheroid Class Reference [Stars and black holes]

Public Member Functions

Protected Member Functions