Transverse and traceless symmetric tensors of rank 2. More...

#include <sym_tensor.h>

Inheritance diagram for Lorene::Sym_tensor_tt:

Public Member Functions
	Sym_tensor_tt (const Map &map, const Base_vect &triad_i, const Metric &met)
	Standard constructor. More...

	Sym_tensor_tt (const Sym_tensor_tt &)
	Copy constructor. More...

	Sym_tensor_tt (const Map &map, const Base_vect &triad_i, const Metric &met, FILE *fich)
	Constructor from a file (see `Tensor::sauve(FILE*)` ). More...

virtual	~Sym_tensor_tt ()
	Destructor. More...

virtual void	operator= (const Sym_tensor_tt &a)
	Assignment to another `Sym_tensor_tt`. More...

virtual void	operator= (const Sym_tensor_trans &a)
	Assignment to a `Sym_tensor_trans`. More...

virtual void	operator= (const Sym_tensor &a)
	Assignment to a `Sym_tensor`. More...

virtual void	operator= (const Tensor_sym &a)
	Assignment to a `Tensor_sym`. More...

virtual void	operator= (const Tensor &a)
	Assignment to a `Tensor`. More...

void	set_rr_eta_mu (const Scalar &hrr, const Scalar &eta_i, const Scalar &mu_i)
	Sets the component $h^{rr}$ , as well as the angular potentials $\eta$ and $\mu$ (see members `p_eta` and `p_mu` ). More...

void	set_rr_mu (const Scalar &hrr, const Scalar &mu_i)
	Sets the component $h^{rr}$ , as well as the angular potential $\mu$ (see member `p_mu` ). More...

void	set_khi_eta_mu (const Scalar &khi_i, const Scalar &eta_i, const Scalar &mu_i)
	Sets the component $\chi$ , as well as the angular potentials $\eta$ and $\mu$ (see members `p_khi` , `p_eta` and `p_mu` ). More...

void	set_khi_mu (const Scalar &khi_i, const Scalar &mu_i, int dzp=0, Param par1=0x0, Param par2=0x0, Param *par3=0x0)
	Sets the component $\chi$ , as well as the angular potential $\mu$ (see member `p_khi` and `p_mu` ). More...

void	set_A_tildeB (const Scalar &a_in, const Scalar &tb_in, Param par_bc=0x0, Param par_mat=0x0)
	Assigns the derived members `A` and $\tilde{B}$ . More...

const Scalar &	khi () const
	Gives the field $\chi$ such that the component $h^{rr} = \frac{\chi}{r^2}$ . More...

virtual const Scalar &	eta (Param *par=0x0) const
	Gives the field $\eta$ (see member `p_eta` ). More...

Sym_tensor_tt	poisson (int dzfin=2) const
	Computes the solution of a tensorial TT Poisson equation with `*this` $= S^{ij}$ as a source: More...

const Metric &	get_met_div () const
	Returns the metric with respect to which the divergence and the trace are defined. More...

void	set_tt_trace (const Sym_tensor_tt &a, const Scalar &h, Param *par=0x0)
	Assigns the derived members `p_tt` and `p_trace` and updates the components accordingly. More...

const Scalar &	the_trace () const
	Returns the trace of the tensor with respect to metric `*met_div`. More...

const Sym_tensor_tt &	tt_part (Param *par=0x0) const
	Returns the transverse traceless part of the tensor, the trace being defined with respect to metric `*met_div`. More...

void	sol_Dirac_Abound (const Scalar &aaa, Scalar &tilde_mu, Scalar &x_new, Scalar bound_mu, const Param *par_bc)
	Same resolution as sol_Dirac_A, but with inner boundary conditions added. More...

void	sol_Dirac_A2 (const Scalar &aaa, Scalar &tilde_mu, Scalar &x_new, Scalar bound_mu, const Param *par_bc)
	Same resolution as sol_Dirac_Abound, but here the boundary conditions are the degenerate elliptic conditions encontered when solving the Kerr problem. More...

void	sol_Dirac_BC2 (const Scalar &bb, const Scalar &cc, const Scalar &hh, Scalar &hrr, Scalar &tilde_eta, Scalar &ww, Scalar bound_eta, double dir, double neum, double rhor, Param par_bc, Param par_mat)
	Same resolution as sol_Dirac_tilde_B, but with inner boundary conditions added. More...

void	sol_Dirac_BC3 (const Scalar &bb, const Scalar &hh, Scalar &hrr, Scalar &tilde_eta, Scalar &ww, Scalar bound_hrr, Scalar bound_eta, Param par_bc, Param par_mat)
	Same resolution as sol_Dirac_Abound, but here the boundary conditions are the degenerate elliptic conditions encontered when solving the Kerr problem. More...

void	sol_Dirac_l01_bound (const Scalar &hh, Scalar &hrr, Scalar &tilde_eta, Scalar &bound_hrr, Scalar &bound_eta, Param *par_mat)

void	sol_Dirac_l01_2 (const Scalar &hh, Scalar &hrr, Scalar &tilde_eta, Param *par_mat)

void	sol_elliptic_ABC (Sym_tensor &source, Scalar aaa, Scalar bbb, Scalar ccc)
	Finds spectral potentials A, B, C of solution of an tensorial TT elliptic equation, given the source. More...

void	trace_from_det_one (const Sym_tensor_tt &htt, double precis=1.e-14, int it_max=100)
	Assigns the derived member `p_tt` and computes the trace so that `*this` + the flat metric has a determinant equal to 1. More...

void	set_hrr_mu_det_one (const Scalar &hrr, const Scalar &mu_in, double precis=1.e-14, int it_max=100)
	Assigns the rr component and the derived member $\mu$ . More...

void	set_tt_part_det_one (const Sym_tensor_tt &hijtt, const Scalar h_prev=0x0, Param par_mat=0x0, double precis=1.e-14, int it_max=100)
	Assignes the TT-part of the tensor. More...

void	set_AtBtt_det_one (const Scalar &a_in, const Scalar &tbtt_in, const Scalar h_prev=0x0, Param par_bc=0x0, Param *par_mat=0x0, double precis=1.e-14, int it_max=100)
	Assigns the derived member `A` and computes $\tilde{B}$ from its TT-part (see `Sym_tensor::compute_tilde_B_tt()` ). More...

void	set_AtB_trace (const Scalar &a_in, const Scalar &tb_in, const Scalar &trace, Param par_bc=0x0, Param par_mat=0x0)
	Assigns the derived members `A` , $\tilde{B}$ and the trace. More...

Sym_tensor_trans	poisson (const Scalar *h_guess=0x0) const
	Computes the solution of a tensorial transverse Poisson equation with `*this` $= S^{ij}$ as a source: More...

void	set_longit_trans (const Vector &v, const Sym_tensor_trans &a)
	Assigns the derived members `p_longit_pot` and `p_transverse` and updates the components accordingly. More...

void	set_auxiliary (const Scalar &trr, const Scalar &eta_over_r, const Scalar &mu_over_r, const Scalar &www, const Scalar &xxx, const Scalar &ttt)
	Assigns the component $T^{rr}$ and the derived members `p_eta` , `p_mu` , `p_www`, `p_xxx` and `p_ttt` , fro, their values and $\eta / r$ , $\mu / r$ . More...

virtual void	exponential_filter_r (int lzmin, int lzmax, int p, double alpha=-16.)
	Applies exponential filters to all components (see `Scalar::exponential_filter_r` ). More...

virtual void	exponential_filter_ylm (int lzmin, int lzmax, int p, double alpha=-16.)
	Applies exponential filters to all components (see `Scalar::exponential_filter_ylm` ). More...

const Vector &	divergence (const Metric &) const
	Returns the divergence of `this` with respect to a `Metric` . More...

Sym_tensor	derive_lie (const Vector &v) const
	Computes the Lie derivative of `this` with respect to some vector field `v`. More...

const Sym_tensor_trans &	transverse (const Metric &gam, Param *par=0x0, int method_poisson=6) const
	Computes the transverse part ${}^t T^{ij}$ of the tensor with respect to a given metric, transverse meaning divergence-free with respect to that metric. More...

const Vector &	longit_pot (const Metric &gam, Param *par=0x0, int method_poisson=6) const
	Computes the vector potential of longitudinal part of the tensor (see documentation of method `transverse()` above). More...

const Scalar &	mu (Param *par=0x0) const
	Gives the field $\mu$ (see member `p_mu` ). More...

const Scalar &	www () const
	Gives the field W (see member `p_www` ). More...

const Scalar &	xxx () const
	Gives the field X (see member `p_xxx` ). More...

const Scalar &	ttt () const
	Gives the field T (see member `p_ttt` ). More...

const Scalar &	compute_A (bool output_ylm=true, Param *par=0x0) const
	Gives the field A (see member `p_aaa` ). More...

const Scalar &	compute_tilde_B (bool output_ylm=true, Param *par=0x0) const
	Gives the field $\tilde{B}$ (see member `p_tilde_b` ). More...

Scalar	compute_tilde_B_tt (bool output_ylm=true, Param *par=0x0) const
	Gives the field $\tilde{B}$ (see member `p_tilde_b` ) associated with the TT-part of the `Sym_tensor` . More...

const Scalar &	compute_tilde_C (bool output_ylm=true, Param *par=0x0) const
	Gives the field $\tilde{C}$ (see member `p_tilde_c` ). More...

int	sym_index1 () const
	Number of the first symmetric index (`0<=` `id_sym1` < `valence` ) More...

int	sym_index2 () const
	Number of the second symmetric index (`id_sym1` < `id_sym2` < `valence` ) More...

virtual int	position (const Itbl &ind) const
	Returns the position in the array `cmp` of a component given by its indices. More...

virtual Itbl	indices (int pos) const
	Returns the indices of a component given by its position in the array `cmp` . More...

virtual void	sauve (FILE *) const
	Save in a binary file. More...

const Tensor_sym &	derive_cov (const Metric &gam) const
	Returns the covariant derivative of `this` with respect to some metric $\gamma$ . More...

const Tensor_sym &	derive_con (const Metric &gam) const
	Returns the "contravariant" derivative of `this` with respect to some metric $\gamma$ , by raising the last index of the covariant derivative (cf. More...

virtual void	set_etat_nondef ()
	Sets the logical state of all components to `ETATNONDEF` (undefined state). More...

virtual void	set_etat_zero ()
	Sets the logical state of all components to `ETATZERO` (zero state). More...

virtual void	set_etat_qcq ()
	Sets the logical state of all components to `ETATQCQ` (ordinary state). More...

virtual void	allocate_all ()
	Performs the memory allocation of all the elements, down to the `double` arrays of the `Tbl` s. More...

virtual void	change_triad (const Base_vect &new_triad)
	Sets a new vectorial basis (triad) of decomposition and modifies the components accordingly. More...

void	set_triad (const Base_vect &new_triad)
	Assigns a new vectorial basis (triad) of decomposition. More...

Scalar &	set (const Itbl &ind)
	Returns the value of a component (read/write version). More...

Scalar &	set (int i1, int i2)
	Returns the value of a component for a tensor of valence 2 (read/write version). More...

Scalar &	set (int i1, int i2, int i3)
	Returns the value of a component for a tensor of valence 3 (read/write version). More...

Scalar &	set (int i1, int i2, int i3, int i4)
	Returns the value of a component for a tensor of valence 4 (read/write version). More...

void	annule_domain (int l)
	Sets the `Tensor` to zero in a given domain. More...

virtual void	annule (int l_min, int l_max)
	Sets the `Tensor` to zero in several domains. More...

void	annule_extern_cn (int l_0, int deg)
	Performs a smooth (C^n) transition in a given domain to zero. More...

virtual void	std_spectral_base ()
	Sets the standard spectal bases of decomposition for each component. More...

virtual void	std_spectral_base_odd ()
	Sets the standard odd spectal bases of decomposition for each component. More...

virtual void	dec_dzpuis (int dec=1)
	Decreases by `dec` units the value of `dzpuis` and changes accordingly the values in the compactified external domain (CED). More...

virtual void	inc_dzpuis (int inc=1)
	Increases by `inc` units the value of `dzpuis` and changes accordingly the values in the compactified external domain (CED). More...

Tensor	up (int ind, const Metric &gam) const
	Computes a new tensor by raising an index of `*this`. More...

Tensor	down (int ind, const Metric &gam) const
	Computes a new tensor by lowering an index of `*this`. More...

Tensor	up_down (const Metric &gam) const
	Computes a new tensor by raising or lowering all the indices of `*this` . More...

Tensor	trace (int ind1, int ind2) const
	Trace on two different type indices. More...

Tensor	trace (int ind1, int ind2, const Metric &gam) const
	Trace with respect to a given metric. More...

Scalar	trace () const
	Trace on two different type indices for a valence 2 tensor. More...

Scalar	trace (const Metric &gam) const
	Trace with respect to a given metric for a valence 2 tensor. More...

const Map &	get_mp () const
	Returns the mapping. More...

const Base_vect *	get_triad () const
	Returns the vectorial basis (triad) on which the components are defined. More...

int	get_valence () const
	Returns the valence. More...

int	get_n_comp () const
	Returns the number of stored components. More...

int	get_index_type (int i) const
	Gives the type (covariant or contravariant) of the index number `i` . More...

Itbl	get_index_type () const
	Returns the types of all the indices. More...

int &	set_index_type (int i)
	Sets the type of the index number `i` . More...

Itbl &	set_index_type ()
	Sets the types of all the indices. More...

const Scalar &	operator() (const Itbl &ind) const
	Returns the value of a component (read-only version). More...

const Scalar &	operator() (int i1, int i2) const
	Returns the value of a component for a tensor of valence 2 (read-only version). More...

const Scalar &	operator() (int i1, int i2, int i3) const
	Returns the value of a component for a tensor of valence 3 (read-only version). More...

const Scalar &	operator() (int i1, int i2, int i3, int i4) const
	Returns the value of a component for a tensor of valence 4 (read-only version). More...

void	operator+= (const Tensor &)
	+= Tensor More...

void	operator-= (const Tensor &)
	-= Tensor More...

virtual void	spectral_display (const char *comment=0x0, double threshold=1.e-7, int precision=4, ostream &ostr=cout) const
	Displays the spectral coefficients and the associated basis functions of each component. More...

Protected Member Functions
virtual void	del_deriv () const
	Deletes the derived quantities. More...

void	set_der_0x0 () const
	Sets the pointers on derived quantities to 0x0. More...

void	update (int dzp, Param par1=0x0, Param par2=0x0)
	Computes the components $h^{r\theta}$ , $h^{r\varphi}$ , $h^{\theta\theta}$ , $h^{\theta\varphi}$ and $h^{\varphi\varphi}$ , from $h^{rr}$ and the potentials $\eta$ and $\mu$ . More...

void	sol_Dirac_A (const Scalar &aaa, Scalar &tilde_mu, Scalar &xxx, const Param *par_bc=0x0) const
	Solves a system of two coupled first-order PDEs obtained from the divergence-free condition (Dirac gauge) and the requirement that the potential A (see `Sym_tensor::p_aaa` ) has a given value. More...

void	sol_Dirac_tilde_B (const Scalar &tilde_b, const Scalar &hh, Scalar &hrr, Scalar &tilde_eta, Scalar &www, Param par_bc=0x0, Param par_mat=0x0) const
	Solves a system of three coupled first-order PDEs obtained from divergence-free conditions (Dirac gauge) and the requirement that the potential $\tilde{B}$ (see `Sym_tensor::p_tilde_b` ) has a given value. More...

void	sol_Dirac_l01 (const Scalar &hh, Scalar &hrr, Scalar &tilde_eta, Param *par_mat) const
	Solves the same system as `Sym_tensor_trans::sol_Dirac_tilde_B` but only for $\ell=0,1$ . More...

virtual void	del_derive_met (int i) const
	Logical destructor of the derivatives depending on the i-th element of `met_depend` specific to the class `Sym_tensor` (`p_transverse` , etc...). More...

void	set_der_met_0x0 (int i) const
	Sets all the i-th components of `met_depend` specific to the class `Sym_tensor` (`p_transverse` , etc...) to 0x0. More...

Scalar	get_tilde_B_from_TT_trace (const Scalar &tilde_B_tt_in, const Scalar &trace) const
	Computes $\tilde{B}$ (see `Sym_tensor::p_tilde_b` ) from its transverse-traceless part and the trace. More...

Sym_tensor *	inverse () const
	Returns a pointer on the inverse of the `Sym_tensor` (seen as a matrix). More...

void	set_dependance (const Metric &) const
	To be used to describe the fact that the derivatives members have been calculated with `met` . More...

int	get_place_met (const Metric &) const
	Returns the position of the pointer on `metre` in the array `met_depend` . More...

void	compute_derive_lie (const Vector &v, Tensor &resu) const
	Computes the Lie derivative of `this` with respect to some vector field `v` (protected method; the public interface is method `derive_lie` ). More...

Protected Attributes
Scalar *	p_khi
	Field $\chi$ such that the component $h^{rr} = \frac{\chi}{r^2}$ . More...

const Metric *const	met_div
	Metric with respect to which the divergence and the trace are defined. More...

Scalar *	p_trace
	Trace with respect to the metric `*met_div` More...

Sym_tensor_tt *	p_tt
	Traceless part with respect to the metric `*met_div` More...

Sym_tensor_trans *	p_transverse [N_MET_MAX]
	Array of the transverse part ${}^t T^{ij}$ of the tensor with respect to various metrics, transverse meaning divergence-free with respect to a metric. More...

Vector *	p_longit_pot [N_MET_MAX]
	Array of the vector potential of the longitudinal part of the tensor with respect to various metrics (see documentation of member `p_transverse`. More...

Scalar *	p_eta
	Field $\eta$ such that the components $(T^{r\theta}, T^{r\varphi})$ of the tensor are written (has only meaning with spherical components!): More...

Scalar *	p_mu
	Field $\mu$ such that the components $(T^{r\theta}, T^{r\varphi})$ of the tensor are written (has only meaning with spherical components!): More...

Scalar *	p_www
	Field W such that the components $T^{\theta\theta}, T^{\varphi\varphi}$ and $T^{\theta\varphi}$ of the tensor are written (has only meaning with spherical components!): More...

Scalar *	p_xxx
	Field X such that the components $T^{\theta\theta}, T^{\varphi\varphi}$ and $T^{\theta\varphi}$ of the tensor are written (has only meaning with spherical components!): More...

Scalar *	p_ttt
	Field T defined as $T = T^{\theta\theta} + T^{\varphi\varphi}$ . More...

Scalar *	p_aaa
	Field A defined from X and $\mu$ insensitive to the longitudinal part of the `Sym_tensor` (only for $\ell \geq 2$ ). More...

Scalar *	p_tilde_b
	Field $\tilde{B}$ defined from $h^{rr}, \eta, W$ and h insensitive to the longitudinal part of the `Sym_tensor`. More...

Scalar *	p_tilde_c
	Field $\tilde{C}$ defined from $h^{rr}, \eta, W$ and h insensitive to the longitudinal part of the `Sym_tensor`. More...

int	id_sym1
	Number of the first symmetric index (`0<=` `id_sym1` < `valence` ) More...

int	id_sym2
	Number of the second symmetric index (`id_sym1` < `id_sym2` < `valence` ) More...

const Map *const	mp
	Mapping on which the numerical values at the grid points are defined. More...

int	valence
	Valence of the tensor (0 = scalar, 1 = vector, etc...) More...

const Base_vect *	triad
	Vectorial basis (triad) with respect to which the tensor components are defined. More...

Itbl	type_indice
	1D array of integers (class `Itbl` ) of size `valence` containing the type of each index: `COV` for a covariant one and `CON` for a contravariant one. More...

int	n_comp
	Number of stored components, depending on the symmetry. More...

Scalar **	cmp
	Array of size `n_comp` of pointers onto the components. More...

const Metric *	met_depend [N_MET_MAX]
	Array on the `Metric` 's which were used to compute derived quantities, like `p_derive_cov` , etc... More...

Tensor *	p_derive_cov [N_MET_MAX]
	Array of pointers on the covariant derivatives of `this` with respect to various metrics. More...

Tensor *	p_derive_con [N_MET_MAX]
	Array of pointers on the contravariant derivatives of `this` with respect to various metrics. More...

Tensor *	p_divergence [N_MET_MAX]
	Array of pointers on the divergence of `this` with respect to various metrics. More...

Detailed Description

Transverse and traceless symmetric tensors of rank 2.

This class is designed to store transverse (divergence-free) and transverse symmetric contravariant tensors of rank 2, with the component expressed in an orthonormal spherical basis $(e_r,e_\theta,e_\varphi)$ .()

Definition at line 938 of file sym_tensor.h.

Constructor & Destructor Documentation

◆ Sym_tensor_tt() [1/3]

Lorene::Sym_tensor_tt::Sym_tensor_tt	(	const Map &	map,
		const Base_vect &	triad_i,
		const Metric &	met
	)

Standard constructor.

Parameters

map	the mapping
triad_i	vectorial basis (triad) with respect to which the tensor components are defined
met	the metric with respect to which the divergence is defined

Definition at line 75 of file sym_tensor_tt.C.

References set_der_0x0().

◆ Sym_tensor_tt() [2/3]

Lorene::Sym_tensor_tt::Sym_tensor_tt ( const Sym_tensor_tt & source )

Copy constructor.

Definition at line 85 of file sym_tensor_tt.C.

References p_khi, and set_der_0x0().

◆ Sym_tensor_tt() [3/3]

Lorene::Sym_tensor_tt::Sym_tensor_tt	(	const Map &	map,
		const Base_vect &	triad_i,
		const Metric &	met,
		FILE *	fich
	)

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

Parameters

map	the mapping
triad_i	vectorial basis (triad) with respect to which the tensor components are defined. It will be checked that it coincides with the basis saved in the file.
met	the metric with respect to which the divergence is defined
fich	file which has been used by the function `sauve(FILE*)` .

Definition at line 97 of file sym_tensor_tt.C.

References set_der_0x0().

◆ ~Sym_tensor_tt()

Lorene::Sym_tensor_tt::~Sym_tensor_tt ( )

virtual

Destructor.

Definition at line 108 of file sym_tensor_tt.C.

References del_deriv().

Member Function Documentation

◆ compute_A()

const Scalar & Lorene::Sym_tensor::compute_A	(	bool	output_ylm = `true`,
		Param *	par = `0x0`
	)		const

inherited

Gives the field A (see member p_aaa ).

Parameters

output_ylm a flag to control the spectral decomposition base of the result: if true (default) the spherical harmonics base is used.

Definition at line 316 of file sym_tensor_aux.C.

References Lorene::Scalar::annule_l(), Lorene::Scalar::div_r_dzpuis(), Lorene::Scalar::div_tant(), Lorene::Scalar::dsdt(), Lorene::Scalar::get_dzpuis(), Lorene::Tensor::mp, Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_aaa, Lorene::Map::poisson_angu(), Lorene::Scalar::poisson_angu(), Lorene::Scalar::set_spectral_va(), Lorene::Scalar::stdsdp(), Lorene::Tensor::triad, Lorene::Sym_tensor::xxx(), Lorene::Valeur::ylm(), and Lorene::Valeur::ylm_i().

◆ compute_tilde_B()

const Scalar & Lorene::Sym_tensor::compute_tilde_B	(	bool	output_ylm = `true`,
		Param *	par = `0x0`
	)		const

inherited

Gives the field $\tilde{B}$ (see member p_tilde_b ).

Parameters

output_ylm a flag to control the spectral decomposition base of the result: if true (default) the spherical harmonics base is used.

Definition at line 362 of file sym_tensor_aux.C.

◆ compute_tilde_B_tt()

Scalar Lorene::Sym_tensor::compute_tilde_B_tt	(	bool	output_ylm = `true`,
		Param *	par = `0x0`
	)		const

inherited

Gives the field $\tilde{B}$ (see member p_tilde_b ) associated with the TT-part of the Sym_tensor .

Parameters

output_ylm a flag to control the spectral decomposition base of the result: if true (default) the spherical harmonics base is used.

Definition at line 478 of file sym_tensor_aux.C.

References Lorene::Sym_tensor::compute_tilde_B(), and Lorene::Scalar::get_etat().

◆ compute_tilde_C()

const Scalar & Lorene::Sym_tensor::compute_tilde_C	(	bool	output_ylm = `true`,
		Param *	par = `0x0`
	)		const

inherited

Gives the field $\tilde{C}$ (see member p_tilde_c ).

Parameters

output_ylm a flag to control the spectral decomposition base of the result: if true (default) the spherical harmonics base is used.

Definition at line 596 of file sym_tensor_aux.C.

◆ del_deriv()

void Lorene::Sym_tensor_tt::del_deriv ( ) const

protectedvirtual

Deletes the derived quantities.

Reimplemented from Lorene::Sym_tensor_trans.

Definition at line 121 of file sym_tensor_tt.C.

References Lorene::Sym_tensor_trans::del_deriv(), p_khi, and set_der_0x0().

◆ del_derive_met()

void Lorene::Sym_tensor::del_derive_met ( int i ) const

protectedvirtualinherited

Logical destructor of the derivatives depending on the i-th element of met_depend specific to the class Sym_tensor (p_transverse , etc...).

Reimplemented from Lorene::Tensor.

Definition at line 320 of file sym_tensor.C.

References Lorene::Tensor::del_derive_met(), Lorene::Tensor::met_depend, Lorene::Sym_tensor::p_longit_pot, Lorene::Sym_tensor::p_transverse, and Lorene::Sym_tensor::set_der_met_0x0().

◆ derive_lie()

Sym_tensor Lorene::Sym_tensor::derive_lie ( const Vector & v ) const

inherited

Computes the Lie derivative of this with respect to some vector field v.

Definition at line 360 of file sym_tensor.C.

References Lorene::Tensor::compute_derive_lie(), Lorene::Tensor::mp, Lorene::Tensor::triad, and Lorene::Tensor::type_indice.

◆ divergence()

const Vector & Lorene::Sym_tensor::divergence ( const Metric & gam ) const

inherited

Returns the divergence of this with respect to a Metric .

The indices are assumed to be contravariant.

Definition at line 349 of file sym_tensor.C.

References Lorene::Tensor::divergence().

◆ eta()

const Scalar & Lorene::Sym_tensor_tt::eta ( Param * par = 0x0 ) const

virtual

Gives the field $\eta$ (see member p_eta ).

Reimplemented from Lorene::Sym_tensor.

Definition at line 140 of file sym_tensor_tt_etamu.C.

References Lorene::Scalar::dec_dzpuis(), Lorene::Scalar::div_r_dzpuis(), Lorene::Scalar::dsdr(), Lorene::Scalar::get_dzpuis(), Lorene::Tensor::mp, Lorene::Scalar::mult_r_dzpuis(), Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_eta, p_khi, Lorene::Map::poisson_angu(), Lorene::Scalar::poisson_angu(), and Lorene::Tensor::triad.

◆ exponential_filter_r()

void Lorene::Sym_tensor::exponential_filter_r	(	int	lzmin,
		int	lzmax,
		int	p,
		double	alpha = `-16.`
	)

virtualinherited

Applies exponential filters to all components (see Scalar::exponential_filter_r ).

Does a loop for Cartesian components, and works in terms of the rr-component, $\eta$ , $\mu$ , W, X, T for spherical components.

Reimplemented from Lorene::Tensor.

Definition at line 446 of file sym_tensor.C.

References Lorene::Tensor::cmp, Lorene::Scalar::div_r(), Lorene::Sym_tensor::eta(), Lorene::Scalar::exponential_filter_r(), Lorene::Map::get_bvect_cart(), Lorene::Map::get_bvect_spher(), Lorene::Base_vect::identify(), Lorene::Tensor::mp, Lorene::Sym_tensor::mu(), Lorene::Tensor::n_comp, Lorene::Tensor::operator()(), Lorene::Sym_tensor::set_auxiliary(), Lorene::Tensor::triad, Lorene::Sym_tensor::ttt(), Lorene::Sym_tensor::www(), and Lorene::Sym_tensor::xxx().

◆ exponential_filter_ylm()

void Lorene::Sym_tensor::exponential_filter_ylm	(	int	lzmin,
		int	lzmax,
		int	p,
		double	alpha = `-16.`
	)

virtualinherited

Applies exponential filters to all components (see Scalar::exponential_filter_ylm ).

Does a loop for Cartesian components, and works in terms of the r-component, $\eta$ , $\mu$ , W, X, T for spherical components.

Reimplemented from Lorene::Tensor.

Definition at line 471 of file sym_tensor.C.

References Lorene::Tensor::cmp, Lorene::Scalar::div_r(), Lorene::Sym_tensor::eta(), Lorene::Scalar::exponential_filter_ylm(), Lorene::Map::get_bvect_cart(), Lorene::Map::get_bvect_spher(), Lorene::Base_vect::identify(), Lorene::Tensor::mp, Lorene::Sym_tensor::mu(), Lorene::Tensor::n_comp, Lorene::Tensor::operator()(), Lorene::Sym_tensor::set_auxiliary(), Lorene::Tensor::triad, Lorene::Sym_tensor::ttt(), Lorene::Sym_tensor::www(), and Lorene::Sym_tensor::xxx().

◆ get_met_div()

const Metric& Lorene::Sym_tensor_trans::get_met_div ( ) const

inlineinherited

Returns the metric with respect to which the divergence and the trace are defined.

Definition at line 669 of file sym_tensor.h.

References Lorene::Sym_tensor_trans::met_div.

◆ get_tilde_B_from_TT_trace()

Scalar Lorene::Sym_tensor::get_tilde_B_from_TT_trace	(	const Scalar &	tilde_B_tt_in,
		const Scalar &	trace
	)		const

protectedinherited

Computes $\tilde{B}$ (see Sym_tensor::p_tilde_b ) from its transverse-traceless part and the trace.

Definition at line 531 of file sym_tensor_aux.C.

References Lorene::Scalar::get_etat(), and Lorene::Tensor::triad.

◆ inverse()

Sym_tensor * Lorene::Sym_tensor::inverse ( ) const

protectedinherited

Returns a pointer on the inverse of the Sym_tensor
(seen as a matrix).

Definition at line 372 of file sym_tensor.C.

References Lorene::Tensor::mp, Lorene::Tensor::operator()(), Lorene::Tensor::set(), Lorene::Tensor::triad, and Lorene::Tensor::type_indice.

◆ khi()

const Scalar & Lorene::Sym_tensor_tt::khi ( ) const

Gives the field $\chi$ such that the component $h^{rr} = \frac{\chi}{r^2}$ .

Definition at line 116 of file sym_tensor_tt_etamu.C.

References Lorene::Scalar::mult_r(), p_khi, and Lorene::Tensor::triad.

◆ longit_pot()

const Vector & Lorene::Sym_tensor::longit_pot	(	const Metric &	gam,
		Param *	par = `0x0`,
		int	method_poisson = `6`
	)		const

inherited

Computes the vector potential $W^i$ of longitudinal part of the tensor (see documentation of method transverse() above).

Parameters

gam	metric with respect to the transverse decomposition is performed
par	parameters for the vector Poisson equation
method_poisson	type of method for solving the vector Poisson equation to get the longitudinal part (see method `Vector::poisson`)

Definition at line 143 of file sym_tensor_decomp.C.

References Lorene::Tensor::dec_dzpuis(), Lorene::Tensor::derive_con(), Lorene::Tensor_sym::derive_con(), Lorene::diffrel(), Lorene::Sym_tensor::divergence(), Lorene::Vector::divergence(), Lorene::Tensor::divergence(), Lorene::Map::get_mg(), Lorene::Mg3d::get_nzone(), Lorene::Tensor::get_place_met(), Lorene::maxabs(), Lorene::Tensor::mp, Lorene::Sym_tensor::p_longit_pot, Lorene::Vector::poisson(), and Lorene::Tensor::set_dependance().

◆ mu()

const Scalar & Lorene::Sym_tensor::mu ( Param * par = 0x0 ) const

inherited

Gives the field $\mu$ (see member p_mu ).

Definition at line 151 of file sym_tensor_aux.C.

References Lorene::Scalar::div_tant(), Lorene::Scalar::dsdt(), Lorene::Scalar::get_dzpuis(), Lorene::Tensor::mp, Lorene::Scalar::mult_r_dzpuis(), Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_mu, Lorene::Map::poisson_angu(), Lorene::Scalar::poisson_angu(), Lorene::Scalar::stdsdp(), and Lorene::Tensor::triad.

◆ operator=() [1/5]

void Lorene::Sym_tensor_tt::operator= ( const Sym_tensor & a )

virtual

Assignment to a Sym_tensor.

Reimplemented from Lorene::Sym_tensor_trans.

Definition at line 163 of file sym_tensor_tt.C.

References del_deriv(), and Lorene::Sym_tensor_trans::operator=().

◆ operator=() [2/5]

void Lorene::Sym_tensor_tt::operator= ( const Sym_tensor_trans & a )

virtual

Assignment to a Sym_tensor_trans.

Reimplemented from Lorene::Sym_tensor_trans.

Definition at line 153 of file sym_tensor_tt.C.

References del_deriv(), and Lorene::Sym_tensor_trans::operator=().

◆ operator=() [3/5]

void Lorene::Sym_tensor_tt::operator= ( const Sym_tensor_tt & a )

virtual

Assignment to another Sym_tensor_tt.

Definition at line 141 of file sym_tensor_tt.C.

References del_deriv(), Lorene::Sym_tensor_trans::operator=(), and p_khi.

◆ operator=() [4/5]

void Lorene::Sym_tensor_tt::operator= ( const Tensor & a )

virtual

Assignment to a Tensor.

Reimplemented from Lorene::Sym_tensor_trans.

Definition at line 181 of file sym_tensor_tt.C.

References del_deriv(), and Lorene::Sym_tensor_trans::operator=().

◆ operator=() [5/5]

void Lorene::Sym_tensor_tt::operator= ( const Tensor_sym & a )

virtual

Assignment to a Tensor_sym.

Reimplemented from Lorene::Sym_tensor_trans.

Definition at line 172 of file sym_tensor_tt.C.

References del_deriv(), and Lorene::Sym_tensor_trans::operator=().

◆ poisson() [1/2]

Sym_tensor_trans Lorene::Sym_tensor_trans::poisson ( const Scalar * h_guess = 0x0 ) const

inherited

Computes the solution of a tensorial transverse Poisson equation with *this $= S^{ij}$ as a source:

$\Delta h^{ij} = S^{ij}. *$

In particular, it makes an iteration on the trace of the result, using Sym_tensor::set_WX_det_one.

Parameters

h_guess a pointer on a guess for the trace of the result; it is passed to Sym_tensor::set_WX_det_one.

Returns: solution $h^{ij}$ of the above equation with the boundary condition $h^{ij}=0$ at spatial infinity.

Definition at line 99 of file sym_tensor_trans_pde.C.

References Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Map::get_mg(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Mg3d::get_type_r(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, and Lorene::Tensor::triad.

◆ poisson() [2/2]

Sym_tensor_tt Lorene::Sym_tensor_tt::poisson ( int dzfin = 2 ) const

Computes the solution of a tensorial TT Poisson equation with *this $= S^{ij}$ as a source:

$\Delta h^{ij} = S^{ij} *$

Parameters

dzfin [input] the dzpuis for all the components of the result (see the documentation for Scalar ).

Returns: solution $h^{ij}$ of the above equation with the boundary condition $h^{ij}=0$ at spatial infinity.

Definition at line 62 of file sym_ttt_poisson.C.

References Lorene::Scalar::get_etat(), Lorene::Map::get_mg(), Lorene::Mg3d::get_nzone(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Tensor::operator()(), and Lorene::Tensor::triad.

◆ set_A_tildeB()

void Lorene::Sym_tensor_tt::set_A_tildeB	(	const Scalar &	a_in,
		const Scalar &	tb_in,
		Param *	par_bc = `0x0`,
		Param *	par_mat = `0x0`
	)

Assigns the derived members A and $\tilde{B}$ .

Other derived members are deduced from the divergence-and trace-free conditions.

Parameters

a_in	the `A` potential (see `Sym_tensor::p_aaa` )
tb_in	the $\tilde{B}$ potential (see `Sym_tensor::p_tilde_b` )

Definition at line 487 of file sym_tensor_tt_etamu.C.

References Lorene::Tensor::mp, and Lorene::Sym_tensor_trans::set_AtB_trace().

◆ set_AtB_trace()

void Lorene::Sym_tensor_trans::set_AtB_trace	(	const Scalar &	a_in,
		const Scalar &	tb_in,
		const Scalar &	trace,
		Param *	par_bc = `0x0`,
		Param *	par_mat = `0x0`
	)

inherited

Assigns the derived members A , $\tilde{B}$ and the trace.

Other derived members are deduced from the divergence-free condition.

Parameters

a_in	the `A` potential (see `Sym_tensor::p_aaa` )
tb_in	the $\tilde{B}$ potential (see `Sym_tensor::p_tilde_b` )
trace	the trace of the `Sym_tensor`.

Definition at line 302 of file sym_tensor_trans_aux.C.

References Lorene::Scalar::check_dzpuis(), Lorene::Tensor::mp, Lorene::Sym_tensor::p_aaa, Lorene::Sym_tensor::p_tilde_b, Lorene::Sym_tensor::set_auxiliary(), Lorene::Sym_tensor_trans::sol_Dirac_A(), Lorene::Sym_tensor_trans::sol_Dirac_tilde_B(), and Lorene::Tensor::triad.

◆ set_AtBtt_det_one()

void Lorene::Sym_tensor_trans::set_AtBtt_det_one	(	const Scalar &	a_in,
		const Scalar &	tbtt_in,
		const Scalar *	h_prev = `0x0`,
		Param *	par_bc = `0x0`,
		Param *	par_mat = `0x0`,
		double	precis = `1.e-14`,
		int	it_max = `100`
	)

inherited

Assigns the derived member A and computes $\tilde{B}$ from its TT-part (see Sym_tensor::compute_tilde_B_tt() ).

Other derived members are deduced from the divergence-free condition. Finally, it computes the trace so that *this + the flat metric has a determinant equal to 1. It then updates the components accordingly. This function makes an iteration until the relative difference in the trace between two steps is lower than precis .

Parameters

a_in	the `A` potential (see `Sym_tensor::p_aaa` )
tbtt_in	the TT-part of $\tilde{B}$ potential (see `Sym_tensor::p_tilde_b` and `Sym_tensor::compute_tilde_B_tt()` )
h_prev	a pointer on a guess for the trace of `*this`; if null, then the iteration starts from 0.
precis	relative difference in the trace computation to end the iteration.
it_max	maximal number of iterations.

Definition at line 137 of file sym_tensor_trans_aux.C.

◆ set_auxiliary()

void Lorene::Sym_tensor::set_auxiliary	(	const Scalar &	trr,
		const Scalar &	eta_over_r,
		const Scalar &	mu_over_r,
		const Scalar &	www,
		const Scalar &	xxx,
		const Scalar &	ttt
	)

inherited

Assigns the component $T^{rr}$ and the derived members p_eta , p_mu , p_www, p_xxx and p_ttt , fro, their values and $\eta / r$ , $\mu / r$ .

It updates the other components accordingly.

Definition at line 266 of file sym_tensor_aux.C.

References Lorene::Scalar::check_dzpuis(), Lorene::Sym_tensor::del_deriv(), Lorene::Scalar::dsdt(), Lorene::Scalar::get_dzpuis(), Lorene::Scalar::lapang(), Lorene::Scalar::mult_r_dzpuis(), Lorene::Sym_tensor::p_eta, Lorene::Sym_tensor::p_mu, Lorene::Sym_tensor::p_ttt, Lorene::Sym_tensor::p_www, Lorene::Sym_tensor::p_xxx, Lorene::Tensor::set(), Lorene::Scalar::set_spectral_va(), Lorene::Scalar::stdsdp(), Lorene::Tensor::triad, and Lorene::Valeur::ylm_i().

◆ set_der_0x0()

void Lorene::Sym_tensor_tt::set_der_0x0 ( ) const

protected

Sets the pointers on derived quantities to 0x0.

Definition at line 131 of file sym_tensor_tt.C.

References p_khi.

◆ set_der_met_0x0()

void Lorene::Sym_tensor::set_der_met_0x0 ( int i ) const

protectedinherited

Sets all the i-th components of met_depend specific to the class Sym_tensor (p_transverse , etc...) to 0x0.

Definition at line 335 of file sym_tensor.C.

References Lorene::Sym_tensor::p_longit_pot, and Lorene::Sym_tensor::p_transverse.

◆ set_hrr_mu_det_one()

void Lorene::Sym_tensor_trans::set_hrr_mu_det_one	(	const Scalar &	hrr,
		const Scalar &	mu_in,
		double	precis = `1.e-14`,
		int	it_max = `100`
	)

inherited

Assigns the rr component and the derived member $\mu$ .

Other derived members are deduced from the divergence-free condition. Finally, it computes T (Sym_tensor::p_ttt ) so that *this + the flat metric has a determinant equal to 1. It then updates the components accordingly. This function makes an iteration until the relative difference in T between two steps is lower than precis .

Parameters

hrr	the rr component of the tensor,
mu_in	the $\mu$ potential,
precis	relative difference in the trace computation to end the iteration.
it_max	maximal number of iterations.

Definition at line 117 of file sym_tensor_trans_aux.C.

References Lorene::Scalar::check_dzpuis(), Lorene::Tensor::dec_dzpuis(), Lorene::Tensor::inc_dzpuis(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Sym_tensor::p_mu, set_rr_mu(), Lorene::Sym_tensor_trans::trace_from_det_one(), and Lorene::Tensor::triad.

◆ set_khi_eta_mu()

void Lorene::Sym_tensor_tt::set_khi_eta_mu	(	const Scalar &	khi_i,
		const Scalar &	eta_i,
		const Scalar &	mu_i
	)

Sets the component $\chi$ , as well as the angular potentials $\eta$ and $\mu$ (see members p_khi , p_eta and p_mu ).

The components are updated consistently by a call to the method update() .

Parameters

khi_i	[input] value of $\chi$
eta_i	[input] angular potential $\eta$
mu_i	[input] angular potential $\mu$

Definition at line 260 of file sym_tensor_tt_etamu.C.

References Lorene::Scalar::div_r(), Lorene::Scalar::get_dzpuis(), Lorene::Sym_tensor::p_eta, p_khi, Lorene::Sym_tensor::p_mu, Lorene::Tensor::set(), Lorene::Tensor::triad, and update().

◆ set_khi_mu()

void Lorene::Sym_tensor_tt::set_khi_mu	(	const Scalar &	khi_i,
		const Scalar &	mu_i,
		int	dzp = `0`,
		Param *	par1 = `0x0`,
		Param *	par2 = `0x0`,
		Param *	par3 = `0x0`
	)

Sets the component $\chi$ , as well as the angular potential $\mu$ (see member p_khi and p_mu ).

The angular potential $\eta$ (member p_eta ) is deduced from the divergence free condition. The tensor components are updated consistently by a call to the method update() .

Parameters

khi_i	[input] value of $\chi$
mu_i	[input] angular potential $\mu$
dzp	[input] `dzpuis` parameter of the resulting tensor components

Definition at line 288 of file sym_tensor_tt_etamu.C.

References Lorene::Scalar::check_dzpuis(), Lorene::Scalar::div_r(), Lorene::Scalar::div_r_dzpuis(), eta(), Lorene::Tensor::mp, p_khi, Lorene::Sym_tensor::p_mu, Lorene::Tensor::set(), Lorene::Tensor::triad, and update().

◆ set_longit_trans()

void Lorene::Sym_tensor::set_longit_trans	(	const Vector &	v,
		const Sym_tensor_trans &	a
	)

inherited

Assigns the derived members p_longit_pot and p_transverse and updates the components accordingly.

(see the documentation of these derived members for details)

Definition at line 88 of file sym_tensor_decomp.C.

References Lorene::Tensor::dec_dzpuis(), Lorene::Sym_tensor::del_deriv(), Lorene::Tensor::get_index_type(), Lorene::Sym_tensor_trans::get_met_div(), Lorene::Tensor::get_place_met(), Lorene::Vector::ope_killing(), Lorene::Sym_tensor::p_longit_pot, Lorene::Sym_tensor::p_transverse, and Lorene::Tensor::set_dependance().

◆ set_rr_eta_mu()

void Lorene::Sym_tensor_tt::set_rr_eta_mu	(	const Scalar &	hrr,
		const Scalar &	eta_i,
		const Scalar &	mu_i
	)

Sets the component $h^{rr}$ , as well as the angular potentials $\eta$ and $\mu$ (see members p_eta and p_mu ).

The other components are updated consistently by a call to the method update() .

Parameters

hrr	[input] value of $h^{rr}$
eta_i	[input] angular potential $\eta$
mu_i	[input] angular potential $\mu$

Definition at line 215 of file sym_tensor_tt_etamu.C.

References Lorene::Scalar::get_dzpuis(), Lorene::Sym_tensor::p_eta, Lorene::Sym_tensor::p_mu, Lorene::Tensor::set(), Lorene::Tensor::triad, and update().

◆ set_rr_mu()

void Lorene::Sym_tensor_tt::set_rr_mu	(	const Scalar &	hrr,
		const Scalar &	mu_i
	)

Sets the component $h^{rr}$ , as well as the angular potential $\mu$ (see member p_mu ).

The angular potential $\eta$ (member p_eta ) is deduced from the divergence free condition. The other tensor components are updated consistently by a call to the method update() .

Parameters

hrr	[input] value of $h^{rr}$
mu_i	[input] angular potential $\mu$

Definition at line 238 of file sym_tensor_tt_etamu.C.

References eta(), Lorene::Scalar::get_dzpuis(), Lorene::Sym_tensor::p_mu, Lorene::Tensor::set(), Lorene::Tensor::triad, and update().

◆ set_tt_part_det_one()

void Lorene::Sym_tensor_trans::set_tt_part_det_one	(	const Sym_tensor_tt &	hijtt,
		const Scalar *	h_prev = `0x0`,
		Param *	par_mat = `0x0`,
		double	precis = `1.e-14`,
		int	it_max = `100`
	)

inherited

Assignes the TT-part of the tensor.

The trace is deduced from the divergence-free condition, through the Dirac system on $\tilde{B}$ , so that *this + the flat metric has a determinant equal to 1. It then updates the components accordingly. This function makes an iteration until the relative difference in the trace between two steps is lower than precis .

Parameters

hijtt	the TT part for `this`.
h_prev	a pointer on a guess for the trace of `*this`; if null, then the iteration starts from 0.
precis	relative difference in the trace computation to end the iteration.
it_max	maximal number of iterations.

Definition at line 226 of file sym_tensor_trans_aux.C.

References Lorene::abs(), Lorene::Scalar::div_r(), eta(), Lorene::Scalar::get_spectral_base(), Lorene::Sym_tensor::get_tilde_B_from_TT_trace(), Lorene::Tensor::inc_dzpuis(), Lorene::max(), Lorene::Tensor::mp, Lorene::Sym_tensor::mu(), Lorene::Sym_tensor_trans::p_trace, Lorene::Sym_tensor_trans::p_tt, Lorene::Sym_tensor::set_auxiliary(), Lorene::Scalar::set_etat_zero(), Lorene::Scalar::set_spectral_base(), Lorene::Sym_tensor_trans::sol_Dirac_tilde_B(), Lorene::Tensor::triad, Lorene::Sym_tensor::www(), and Lorene::Sym_tensor::xxx().

◆ set_tt_trace()

void Lorene::Sym_tensor_trans::set_tt_trace	(	const Sym_tensor_tt &	a,
		const Scalar &	h,
		Param *	par = `0x0`
	)

inherited

Assigns the derived members p_tt and p_trace and updates the components accordingly.

(see the documentation of these derived members for details)

Definition at line 235 of file sym_tensor_trans.C.

References Lorene::Scalar::check_dzpuis(), Lorene::Metric::con(), Lorene::Tensor::dec_dzpuis(), Lorene::Sym_tensor_trans::del_deriv(), Lorene::Scalar::derive_con(), Lorene::Tensor_sym::derive_con(), Lorene::Sym_tensor_trans::get_met_div(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Sym_tensor_trans::p_trace, Lorene::Sym_tensor_trans::p_tt, and Lorene::Scalar::poisson().

◆ sol_Dirac_A()

void Lorene::Sym_tensor_trans::sol_Dirac_A	(	const Scalar &	aaa,
		Scalar &	tilde_mu,
		Scalar &	xxx,
		const Param *	par_bc = `0x0`
	)		const

protectedinherited

Solves a system of two coupled first-order PDEs obtained from the divergence-free condition (Dirac gauge) and the requirement that the potential A (see Sym_tensor::p_aaa ) has a given value.

The system reads:

$\begin{eqnarray*} \frac{\partial \tilde{\mu}}{\partial r} + \frac{3\tilde{\mu}}{r} + \left( \Delta_{\theta\varphi } + 2\right) X &=& 0;\\ \frac{\partial X}{\partial r} - \frac{\tilde{\mu}}{r} &=& A. \end{eqnarray*}$

Note that this is solved only for $\ell \geq 2$ and that $\tilde{\mu} = \mu / r$ (see Sym_tensor::p_mu ).

Parameters

aaa	[input] the source A
tilde_mu	[output] the solution $\tilde{\mu}$
xxx	[output] the solution X
par_bc	[input] `Param` to control the boundary conditions

Definition at line 82 of file sym_tensor_trans_dirac.C.

References Lorene::Map::get_mg(), Lorene::Mg3d::get_type_r(), and Lorene::Tensor::mp.

◆ sol_Dirac_A2()

void Lorene::Sym_tensor_trans::sol_Dirac_A2	(	const Scalar &	aaa,
		Scalar &	tilde_mu,
		Scalar &	x_new,
		Scalar	bound_mu,
		const Param *	par_bc
	)

inherited

Same resolution as sol_Dirac_Abound, but here the boundary conditions are the degenerate elliptic conditions encontered when solving the Kerr problem.

Definition at line 83 of file sym_tensor_trans_dirac_boundfree.C.

References Lorene::Map::get_mg(), Lorene::Mg3d::get_type_r(), and Lorene::Tensor::mp.

◆ sol_Dirac_Abound()

void Lorene::Sym_tensor_trans::sol_Dirac_Abound	(	const Scalar &	aaa,
		Scalar &	tilde_mu,
		Scalar &	x_new,
		Scalar	bound_mu,
		const Param *	par_bc
	)

inherited

Same resolution as sol_Dirac_A, but with inner boundary conditions added.

For now, only Robyn-type boundary conditions on $\frac {\mu} {r}$ can be imposed.

Definition at line 81 of file sym_tensor_trans_dirac_bound2.C.

References Lorene::Map::get_mg(), Lorene::Mg3d::get_type_r(), and Lorene::Tensor::mp.

◆ sol_Dirac_BC2()

void Lorene::Sym_tensor_trans::sol_Dirac_BC2	(	const Scalar &	bb,
		const Scalar &	cc,
		const Scalar &	hh,
		Scalar &	hrr,
		Scalar &	tilde_eta,
		Scalar &	ww,
		Scalar	bound_eta,
		double	dir,
		double	neum,
		double	rhor,
		Param *	par_bc,
		Param *	par_mat
	)

inherited

Same resolution as sol_Dirac_tilde_B, but with inner boundary conditions added.

The difference is here, one has to put B and C values in (and not only $\tilde{B}$ ). For now, only Robyn-type boundary conditions on $h^{rr}$ can be imposed.

Definition at line 580 of file sym_tensor_trans_dirac_bound2.C.

References Lorene::Map::get_mg(), Lorene::Mg3d::get_type_r(), and Lorene::Tensor::mp.

◆ sol_Dirac_BC3()

void Lorene::Sym_tensor_trans::sol_Dirac_BC3	(	const Scalar &	bb,
		const Scalar &	hh,
		Scalar &	hrr,
		Scalar &	tilde_eta,
		Scalar &	ww,
		Scalar	bound_hrr,
		Scalar	bound_eta,
		Param *	par_bc,
		Param *	par_mat
	)

inherited

Same resolution as sol_Dirac_Abound, but here the boundary conditions are the degenerate elliptic conditions encontered when solving the Kerr problem.

Definition at line 603 of file sym_tensor_trans_dirac_boundfree.C.

References Lorene::Map::get_mg(), Lorene::Mg3d::get_type_r(), and Lorene::Tensor::mp.

◆ sol_Dirac_l01()

void Lorene::Sym_tensor_trans::sol_Dirac_l01	(	const Scalar &	hh,
		Scalar &	hrr,
		Scalar &	tilde_eta,
		Param *	par_mat
	)		const

protectedinherited

Solves the same system as Sym_tensor_trans::sol_Dirac_tilde_B but only for $\ell=0,1$ .

In these particular cases, W =0 the system is simpler and homogeneous solutions are different.

Definition at line 1438 of file sym_tensor_trans_dirac.C.

References Lorene::Scalar::get_etat(), and Lorene::Tensor::mp.

◆ sol_Dirac_tilde_B()

void Lorene::Sym_tensor_trans::sol_Dirac_tilde_B	(	const Scalar &	tilde_b,
		const Scalar &	hh,
		Scalar &	hrr,
		Scalar &	tilde_eta,
		Scalar &	www,
		Param *	par_bc = `0x0`,
		Param *	par_mat = `0x0`
	)		const

protectedinherited

Solves a system of three coupled first-order PDEs obtained from divergence-free conditions (Dirac gauge) and the requirement that the potential $\tilde{B}$ (see Sym_tensor::p_tilde_b ) has a given value.

The system reads:

$\begin{eqnarray*} \frac{\partial T^{rr}}{r} + \frac{3T^{rr}}{r} +\frac{1}{r} \Delta_{\theta\varphi } \tilde{\eta} &=& \frac{h}{r};\\ \frac{\partial \tilde{\eta}}{\partial r} + \frac{3\tilde{\eta}}{r} - \frac{T^{rr}}{2r} + \left( \Delta_{\theta\varphi } + 2\right) \frac{W}{r} &=& -\frac{h}{2r};\\ (\ell + 2) \frac{\partial W}{\partial r} + \ell(\ell + 2) \frac{W}{r} - \frac{2\tilde{\eta}}{r} + \frac{(\ell +2)T}{2r(\ell + 1)} + \frac{1}{2(\ell + 1)} \frac{\partial T}{\partial r} - \frac{T^{rr}} {(\ell + 1)r} &=& \tilde{B} - \frac{1}{2(\ell +1)} \frac{\partial h} {\partial r} - \frac{\ell +2}{\ell +1} \frac{h}{2r}.\end{eqnarray*}$

Note that $\tilde{\eta} = \eta / r$ (for definitions, see derived members of Sym_tensor).

Parameters

tilde_b	[input] the source $\tilde{B}$
hh	[input] the trace of the tensor
hrr	[output] the rr component of the result
tilde_eta	[output] the solution $\tilde{\eta}$
www	[output] the solution W
par_bc	[input] `Param` to control the boundary conditions
par_mat	[input/output] `Param` in which the operator matrix is stored.

Definition at line 583 of file sym_tensor_trans_dirac.C.

References Lorene::Map::get_mg(), Lorene::Mg3d::get_type_r(), and Lorene::Tensor::mp.

◆ sol_elliptic_ABC()

void Lorene::Sym_tensor_trans::sol_elliptic_ABC	(	Sym_tensor &	source,
		Scalar	aaa,
		Scalar	bbb,
		Scalar	ccc
	)

inherited

Finds spectral potentials A, B, C of solution of an tensorial TT elliptic equation, given the source.

◆ the_trace()

const Scalar & Lorene::Sym_tensor_trans::the_trace ( ) const

inherited

Returns the trace of the tensor with respect to metric *met_div.

Definition at line 270 of file sym_tensor_trans.C.

References Lorene::Sym_tensor_trans::met_div, Lorene::Sym_tensor_trans::p_trace, Lorene::Tensor::trace(), and Lorene::Tensor::type_indice.

◆ trace_from_det_one()

void Lorene::Sym_tensor_trans::trace_from_det_one	(	const Sym_tensor_tt &	htt,
		double	precis = `1.e-14`,
		int	it_max = `100`
	)

inherited

Assigns the derived member p_tt and computes the trace so that *this + the flat metric has a determinant equal to 1.

It then updates the components accordingly, with a dzpuis = 2. This function makes an iteration until the relative difference in the trace between two steps is lower than precis .

Parameters

htt	the transverse traceless part; all components must have dzpuis = 2.
precis	relative difference in the trace computation to end the iteration.
it_max	maximal number of iterations.

Definition at line 315 of file sym_tensor_trans.C.

References Lorene::abs(), Lorene::Scalar::check_dzpuis(), Lorene::Tensor::cmp, Lorene::Scalar::dec_dzpuis(), Lorene::Sym_tensor_trans::get_met_div(), Lorene::Tensor::get_n_comp(), Lorene::max(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Scalar::set_etat_zero(), and Lorene::Sym_tensor_trans::set_tt_trace().

◆ transverse()

const Sym_tensor_trans & Lorene::Sym_tensor::transverse	(	const Metric &	gam,
		Param *	par = `0x0`,
		int	method_poisson = `6`
	)		const

inherited

Computes the transverse part ${}^t T^{ij}$ of the tensor with respect to a given metric, transverse meaning divergence-free with respect to that metric.

Denoting *this by $T^{ij}$ , we then have

$T^{ij} = {}^t T^{ij} + \nabla^i W^j + \nabla^j W^i \qquad\mbox{with}\quad \nabla_j {}^t T^{ij} = 0 *$

where $\nabla_i$ denotes the covariant derivative with respect to the given metric and $W^i$ is the vector potential of the longitudinal part of $T^{ij}$ (function longit_pot() below)

Parameters

gam	metric with respect to the transverse decomposition is performed
par	parameters for the vector Poisson equation
method_poisson	type of method for solving the vector Poisson equation to get the longitudinal part (see method `Vector::poisson`)

Definition at line 110 of file sym_tensor_decomp.C.

References Lorene::Tensor::cmp, Lorene::Tensor::get_place_met(), Lorene::Tensor::inc_dzpuis(), Lorene::Sym_tensor::longit_pot(), Lorene::Tensor::mp, Lorene::Tensor::n_comp, Lorene::Vector::ope_killing(), Lorene::Sym_tensor::p_transverse, Lorene::Tensor::set_dependance(), Lorene::Tensor::triad, and Lorene::Tensor::type_indice.

◆ tt_part()

const Sym_tensor_tt & Lorene::Sym_tensor_trans::tt_part ( Param * par = 0x0 ) const

inherited

Returns the transverse traceless part of the tensor, the trace being defined with respect to metric *met_div.

Definition at line 284 of file sym_tensor_trans.C.

References Lorene::Metric::con(), Lorene::Tensor::dec_dzpuis(), Lorene::Scalar::derive_con(), Lorene::Tensor_sym::derive_con(), Lorene::Scalar::get_dzpuis(), Lorene::Tensor::inc_dzpuis(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Sym_tensor_trans::p_tt, Lorene::Scalar::poisson(), Lorene::Sym_tensor_trans::the_trace(), and Lorene::Tensor::triad.

◆ ttt()

const Scalar & Lorene::Sym_tensor::ttt ( ) const

inherited

Gives the field T (see member p_ttt ).

Definition at line 190 of file sym_tensor_aux.C.

References Lorene::Sym_tensor::p_ttt, and Lorene::Tensor::triad.

◆ update()

void Lorene::Sym_tensor_tt::update	(	int	dzp,
		Param *	par1 = `0x0`,
		Param *	par2 = `0x0`
	)

protected

Computes the components $h^{r\theta}$ , $h^{r\varphi}$ , $h^{\theta\theta}$ , $h^{\theta\varphi}$ and $h^{\varphi\varphi}$ , from $h^{rr}$ and the potentials $\eta$ and $\mu$ .

Parameters

dzp	`dzpuis` parameter of the result, i.e. of the components $h^{ij}$ .

Definition at line 338 of file sym_tensor_tt_etamu.C.

◆ www()

const Scalar & Lorene::Sym_tensor::www ( ) const

inherited

Gives the field W (see member p_www ).

Definition at line 209 of file sym_tensor_aux.C.

References Lorene::Scalar::div_tant(), Lorene::Scalar::dsdt(), Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_www, Lorene::Scalar::poisson_angu(), Lorene::Scalar::stdsdp(), and Lorene::Tensor::triad.

◆ xxx()

const Scalar & Lorene::Sym_tensor::xxx ( ) const

inherited

Gives the field X (see member p_xxx ).

Definition at line 240 of file sym_tensor_aux.C.

References Lorene::Scalar::div_tant(), Lorene::Scalar::dsdt(), Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_xxx, Lorene::Scalar::poisson_angu(), Lorene::Scalar::stdsdp(), and Lorene::Tensor::triad.

Member Data Documentation

◆ met_div

const Metric* const Lorene::Sym_tensor_trans::met_div

protectedinherited

Metric with respect to which the divergence and the trace are defined.

Definition at line 614 of file sym_tensor.h.

◆ p_aaa

Scalar* Lorene::Sym_tensor::p_aaa

mutableprotectedinherited

Field A defined from X and $\mu$ insensitive to the longitudinal part of the Sym_tensor (only for $\ell \geq 2$ ).

Its definition reads

$A = \frac{\partial X}{\partial r} - \frac{\mu}{r^2}.$

Definition at line 322 of file sym_tensor.h.

◆ p_eta

Scalar* Lorene::Sym_tensor::p_eta

mutableprotectedinherited

Field $\eta$ such that the components $(T^{r\theta}, T^{r\varphi})$ of the tensor are written (has only meaning with spherical components!):

$T^{r\theta} = {1\over r} \left( {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \right) *$

$T^{r\varphi} = {1\over r} \left( {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \right) *$

Definition at line 260 of file sym_tensor.h.

◆ p_khi

Scalar* Lorene::Sym_tensor_tt::p_khi

mutableprotected

Field $\chi$ such that the component $h^{rr} = \frac{\chi}{r^2}$ .

Definition at line 946 of file sym_tensor.h.

◆ p_longit_pot

Vector* Lorene::Sym_tensor::p_longit_pot[N_MET_MAX]

mutableprotectedinherited

Array of the vector potential of the longitudinal part of the tensor with respect to various metrics (see documentation of member p_transverse.

Definition at line 246 of file sym_tensor.h.

◆ p_mu

Scalar* Lorene::Sym_tensor::p_mu

mutableprotectedinherited

Field $\mu$ such that the components $(T^{r\theta}, T^{r\varphi})$ of the tensor are written (has only meaning with spherical components!):

$T^{r\theta} = {1\over r} \left( {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \right) *$

$T^{r\varphi} = {1\over r} \left( {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \right) *$

Definition at line 274 of file sym_tensor.h.

◆ p_tilde_b

Scalar* Lorene::Sym_tensor::p_tilde_b

mutableprotectedinherited

Field $\tilde{B}$ defined from $h^{rr}, \eta, W$ and h insensitive to the longitudinal part of the Sym_tensor.

It is defined for each multipolar momentum $\ell \geq 2$ by

$\tilde{B} = (\ell + 2) \frac{\partial W}{\partial r} + \ell(\ell + 2) \frac{W}{r} - \frac{2\eta}{r^2} + \frac{(\ell +2)T}{2r(\ell + 1)} + \frac{1}{2(\ell + 1)} \frac{\partial T}{\partial r} - \frac{h^{rr}} {(\ell + 1)r}.$

Definition at line 334 of file sym_tensor.h.

◆ p_tilde_c

Scalar* Lorene::Sym_tensor::p_tilde_c

mutableprotectedinherited

Field $\tilde{C}$ defined from $h^{rr}, \eta, W$ and h insensitive to the longitudinal part of the Sym_tensor.

It is defined for each multipolar momentum $\ell \geq 2$ by

$\tilde{C} = - (\ell - 1) \frac{\partial W}{\partial r} + (\ell + 1)(\ell - 1) \frac{W}{r} - \frac{2\eta}{r^2} + \frac{(\ell - 1)T}{2r\ell} - \frac{1}{2 \ell } \frac{\partial T}{\partial r} - \frac{h^{rr}} {\ell r}.$

Definition at line 346 of file sym_tensor.h.

◆ p_trace

Scalar* Lorene::Sym_tensor_trans::p_trace

mutableprotectedinherited

Trace with respect to the metric *met_div

Definition at line 617 of file sym_tensor.h.

◆ p_transverse

Sym_tensor_trans* Lorene::Sym_tensor::p_transverse[N_MET_MAX]

mutableprotectedinherited

Array of the transverse part ${}^t T^{ij}$ of the tensor with respect to various metrics, transverse meaning divergence-free with respect to a metric.

Denoting *this by $T^{ij}$ , we then have

$T^{ij} = {}^t T^{ij} + \nabla^i W^j + \nabla^j W^i \qquad\mbox{with}\quad \nabla_j {}^t T^{ij} = 0 *$

where $\nabla_i$ denotes the covariant derivative with respect to the given metric and $W^i$ is the vector potential of the longitudinal part of $T^{ij}$ (member p_longit_pot below)

Definition at line 239 of file sym_tensor.h.

◆ p_tt

Sym_tensor_tt* Lorene::Sym_tensor_trans::p_tt

mutableprotectedinherited

Traceless part with respect to the metric *met_div

Definition at line 620 of file sym_tensor.h.

◆ p_ttt

Scalar* Lorene::Sym_tensor::p_ttt

mutableprotectedinherited

Field T defined as $T = T^{\theta\theta} + T^{\varphi\varphi}$ .

Definition at line 315 of file sym_tensor.h.

◆ p_www

Scalar* Lorene::Sym_tensor::p_www

mutableprotectedinherited

Field W such that the components $T^{\theta\theta}, T^{\varphi\varphi}$ and $T^{\theta\varphi}$ of the tensor are written (has only meaning with spherical components!):

$\frac{1}{2}\left(T^{\theta\theta} - T^{\varphi\varphi} \right) = \frac{\partial^2 W}{\partial\theta^2} - \frac{1}{\tan \theta} \frac{\partial W}{\partial \theta} - \frac{1}{\sin^2 \theta} \frac{\partial^2 W}{\partial \varphi^2} - 2\frac{\partial}{\partial \theta} \left( \frac{1}{\sin \theta} \frac{\partial X}{\partial \varphi} \right) , *$

$T^{\theta\varphi} = \frac{\partial^2 X}{\partial\theta^2} - \frac{1}{\tan \theta} \frac{\partial X}{\partial \theta} - \frac{1}{\sin^2 \theta} \frac{\partial^2 X}{\partial \varphi^2} + 2\frac{\partial}{\partial \theta} \left( \frac{1}{\sin \theta} \frac{\partial W}{\partial \varphi} \right) . *$

Definition at line 293 of file sym_tensor.h.

◆ p_xxx

Scalar* Lorene::Sym_tensor::p_xxx

mutableprotectedinherited

Field X such that the components $T^{\theta\theta}, T^{\varphi\varphi}$ and $T^{\theta\varphi}$ of the tensor are written (has only meaning with spherical components!):

$\frac{1}{2}\left(T^{\theta\theta} - T^{\varphi\varphi} \right) = \frac{\partial^2 W}{\partial\theta^2} - \frac{1}{\tan \theta} \frac{\partial W}{\partial \theta} - \frac{1}{\sin^2 \theta} \frac{\partial^2 W}{\partial \varphi^2} - 2\frac{\partial}{\partial \theta} \left( \frac{1}{\sin \theta} \frac{\partial X}{\partial \varphi} \right) , *$

$T^{\theta\varphi} = \frac{\partial^2 X}{\partial\theta^2} - \frac{1}{\tan \theta} \frac{\partial X}{\partial \theta} - \frac{1}{\sin^2 \theta} \frac{\partial^2 X}{\partial \varphi^2} + 2\frac{\partial}{\partial \theta} \left( \frac{1}{\sin \theta} \frac{\partial W}{\partial \varphi} \right) . *$

Definition at line 312 of file sym_tensor.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

◆ Sym_tensor_tt() [1/3]

◆ Sym_tensor_tt() [2/3]

◆ Sym_tensor_tt() [3/3]

◆ ~Sym_tensor_tt()

Member Function Documentation

◆ compute_A()

◆ compute_tilde_B()

◆ compute_tilde_B_tt()

◆ compute_tilde_C()

◆ del_deriv()

◆ del_derive_met()

◆ derive_lie()

◆ divergence()

◆ eta()

◆ exponential_filter_r()

◆ exponential_filter_ylm()

◆ get_met_div()

◆ get_tilde_B_from_TT_trace()

◆ inverse()

◆ khi()

◆ longit_pot()

◆ mu()

◆ operator=() [1/5]

◆ operator=() [2/5]

◆ operator=() [3/5]

◆ operator=() [4/5]

◆ operator=() [5/5]

◆ poisson() [1/2]

◆ poisson() [2/2]

◆ set_A_tildeB()

◆ set_AtB_trace()

◆ set_AtBtt_det_one()

◆ set_auxiliary()

◆ set_der_0x0()

◆ set_der_met_0x0()

◆ set_hrr_mu_det_one()

◆ set_khi_eta_mu()

◆ set_khi_mu()

◆ set_longit_trans()

◆ set_rr_eta_mu()

◆ set_rr_mu()

◆ set_tt_part_det_one()

◆ set_tt_trace()

◆ sol_Dirac_A()

◆ sol_Dirac_A2()

◆ sol_Dirac_Abound()

◆ sol_Dirac_BC2()

◆ sol_Dirac_BC3()

◆ sol_Dirac_l01()

◆ sol_Dirac_tilde_B()

◆ sol_elliptic_ABC()

◆ the_trace()

◆ trace_from_det_one()

◆ transverse()

◆ tt_part()

◆ ttt()

◆ update()

◆ www()

◆ xxx()

Member Data Documentation

◆ met_div

◆ p_aaa

◆ p_eta

◆ p_khi

◆ p_longit_pot

◆ p_mu

◆ p_tilde_b

◆ p_tilde_c

◆ p_trace

◆ p_transverse

◆ p_tt

◆ p_ttt

◆ p_www

◆ p_xxx