LORENE
Lorene::Sym_tensor_tt Class Reference

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

#include <sym_tensor.h>

Inheritance diagram for Lorene::Sym_tensor_tt:
Lorene::Sym_tensor_trans Lorene::Sym_tensor Lorene::Tensor_sym Lorene::Tensor

Public Member Functions

 Sym_tensor_tt (const Map &map, const Base_vect &triad_i, const Metric &met)
 Standard constructor.
 
 Sym_tensor_tt (const Sym_tensor_tt &)
 Copy constructor.
 
 Sym_tensor_tt (const Map &map, const Base_vect &triad_i, const Metric &met, FILE *fich)
 Constructor from a file (see Tensor::sauve(FILE*) ).
 
virtual ~Sym_tensor_tt ()
 Destructor.
 
virtual void operator= (const Sym_tensor_tt &a)
 Assignment to another Sym_tensor_tt.
 
virtual void operator= (const Sym_tensor_trans &a)
 Assignment to a Sym_tensor_trans.
 
virtual void operator= (const Sym_tensor &a)
 Assignment to a Sym_tensor.
 
virtual void operator= (const Tensor_sym &a)
 Assignment to a Tensor_sym.
 
virtual void operator= (const Tensor &a)
 Assignment to a Tensor.
 
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 ).
 
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 ).
 
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 ).
 
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 ).
 
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 $\tilde{B}$.
 
const Scalarkhi () const
 Gives the field $\chi$ such that the component $h^{rr} = \frac{\chi}{r^2}$.
 
virtual const Scalareta (Param *par=0x0) const
 Gives the field $\eta$ (see member p_eta ).
 
Sym_tensor_tt poisson (int dzfin=2) const
 Computes the solution of a tensorial TT Poisson equation with *this $= S^{ij}$ as a source:
 
const Metricget_met_div () const
 Returns the metric with respect to which the divergence and the trace are defined.
 
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.
 
const Scalarthe_trace () const
 Returns the trace of the tensor with respect to metric *met_div.
 
const Sym_tensor_tttt_part (Param *par=0x0) const
 Returns the transverse traceless part of the tensor, the trace being defined with respect to metric *met_div.
 
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.
 
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.
 
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.
 
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.
 
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.
 
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.
 
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 $\mu$.
 
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.
 
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() ).
 
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.
 
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:
 
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.
 
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 $.
 
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 ).
 
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 ).
 
const Vectordivergence (const Metric &) const
 Returns the divergence of this with respect to a Metric .
 
Sym_tensor derive_lie (const Vector &v) const
 Computes the Lie derivative of this with respect to some vector field v.
 
const Sym_tensor_transtransverse (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.
 
const Vectorlongit_pot (const Metric &gam, Param *par=0x0, int method_poisson=6) const
 Computes the vector potential $W^i$ of longitudinal part of the tensor (see documentation of method transverse() above).
 
const Scalarmu (Param *par=0x0) const
 Gives the field $\mu$ (see member p_mu ).
 
const Scalarwww () const
 Gives the field W (see member p_www ).
 
const Scalarxxx () const
 Gives the field X (see member p_xxx ).
 
const Scalarttt () const
 Gives the field T (see member p_ttt ).
 
const Scalarcompute_A (bool output_ylm=true, Param *par=0x0) const
 Gives the field A (see member p_aaa ).
 
const Scalarcompute_tilde_B (bool output_ylm=true, Param *par=0x0) const
 Gives the field $\tilde{B}$ (see member p_tilde_b ).
 
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 .
 
const Scalarcompute_tilde_C (bool output_ylm=true, Param *par=0x0) const
 Gives the field $\tilde{C}$ (see member p_tilde_c ).
 
int sym_index1 () const
 Number of the first symmetric index (0<= id_sym1 < valence )
 
int sym_index2 () const
 Number of the second symmetric index (id_sym1 < id_sym2 < valence )
 
virtual int position (const Itbl &ind) const
 Returns the position in the array cmp of a component given by its indices.
 
virtual Itbl indices (int pos) const
 Returns the indices of a component given by its position in the array cmp .
 
virtual void sauve (FILE *) const
 Save in a binary file.
 
const Tensor_symderive_cov (const Metric &gam) const
 Returns the covariant derivative of this with respect to some metric $\gamma$.
 
const Tensor_symderive_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.
 
virtual void set_etat_nondef ()
 Sets the logical state of all components to ETATNONDEF
(undefined state).
 
virtual void set_etat_zero ()
 Sets the logical state of all components to ETATZERO
(zero state).
 
virtual void set_etat_qcq ()
 Sets the logical state of all components to ETATQCQ
(ordinary state).
 
virtual void allocate_all ()
 Performs the memory allocation of all the elements, down to the double arrays of the Tbl s.
 
virtual void change_triad (const Base_vect &new_triad)
 Sets a new vectorial basis (triad) of decomposition and modifies the components accordingly.
 
void set_triad (const Base_vect &new_triad)
 Assigns a new vectorial basis (triad) of decomposition.
 
Scalarset (const Itbl &ind)
 Returns the value of a component (read/write version).
 
Scalarset (int i1, int i2)
 Returns the value of a component for a tensor of valence 2 (read/write version).
 
Scalarset (int i1, int i2, int i3)
 Returns the value of a component for a tensor of valence 3 (read/write version).
 
Scalarset (int i1, int i2, int i3, int i4)
 Returns the value of a component for a tensor of valence 4 (read/write version).
 
void annule_domain (int l)
 Sets the Tensor to zero in a given domain.
 
virtual void annule (int l_min, int l_max)
 Sets the Tensor to zero in several domains.
 
void annule_extern_cn (int l_0, int deg)
 Performs a smooth (C^n) transition in a given domain to zero.
 
virtual void std_spectral_base ()
 Sets the standard spectal bases of decomposition for each component.
 
virtual void std_spectral_base_odd ()
 Sets the standard odd spectal bases of decomposition for each component.
 
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).
 
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).
 
Tensor up (int ind, const Metric &gam) const
 Computes a new tensor by raising an index of *this.
 
Tensor down (int ind, const Metric &gam) const
 Computes a new tensor by lowering an index of *this.
 
Tensor up_down (const Metric &gam) const
 Computes a new tensor by raising or lowering all the indices of *this .
 
Tensor trace (int ind1, int ind2) const
 Trace on two different type indices.
 
Tensor trace (int ind1, int ind2, const Metric &gam) const
 Trace with respect to a given metric.
 
Scalar trace () const
 Trace on two different type indices for a valence 2 tensor.
 
Scalar trace (const Metric &gam) const
 Trace with respect to a given metric for a valence 2 tensor.
 
const Mapget_mp () const
 Returns the mapping.
 
const Base_vectget_triad () const
 Returns the vectorial basis (triad) on which the components are defined.
 
int get_valence () const
 Returns the valence.
 
int get_n_comp () const
 Returns the number of stored components.
 
int get_index_type (int i) const
 Gives the type (covariant or contravariant) of the index number i .
 
Itbl get_index_type () const
 Returns the types of all the indices.
 
intset_index_type (int i)
 Sets the type of the index number i .
 
Itblset_index_type ()
 Sets the types of all the indices.
 
const Scalaroperator() (const Itbl &ind) const
 Returns the value of a component (read-only version).
 
const Scalaroperator() (int i1, int i2) const
 Returns the value of a component for a tensor of valence 2 (read-only version).
 
const Scalaroperator() (int i1, int i2, int i3) const
 Returns the value of a component for a tensor of valence 3 (read-only version).
 
const Scalaroperator() (int i1, int i2, int i3, int i4) const
 Returns the value of a component for a tensor of valence 4 (read-only version).
 
void operator+= (const Tensor &)
 += Tensor
 
void operator-= (const Tensor &)
 -= Tensor
 
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.
 

Protected Member Functions

virtual void del_deriv () const
 Deletes the derived quantities.
 
void set_der_0x0 () const
 Sets the pointers on derived quantities to 0x0.
 
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 $\mu$.
 
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.
 
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.
 
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 $\ell=0,1$.
 
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...).
 
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.
 
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.
 
Sym_tensorinverse () const
 Returns a pointer on the inverse of the Sym_tensor
(seen as a matrix).
 
void set_dependance (const Metric &) const
 To be used to describe the fact that the derivatives members have been calculated with met .
 
int get_place_met (const Metric &) const
 Returns the position of the pointer on metre in the array met_depend .
 
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 ).
 

Protected Attributes

Scalarp_khi
 Field $\chi$ such that the component $h^{rr} = \frac{\chi}{r^2}$.
 
const Metric *const met_div
 Metric with respect to which the divergence and the trace are defined.
 
Scalarp_trace
 Trace with respect to the metric *met_div

 
Sym_tensor_ttp_tt
 Traceless part with respect to the metric *met_div

 
Sym_tensor_transp_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.
 
Vectorp_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.
 
Scalarp_eta
 Field $\eta$ such that the components $(T^{r\theta}, T^{r\varphi})$ of the tensor are written (has only meaning with spherical components!):
 
Scalarp_mu
 Field $\mu$ such that the components $(T^{r\theta}, T^{r\varphi})$ of the tensor are written (has only meaning with spherical components!):
 
Scalarp_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!):
 
Scalarp_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!):
 
Scalarp_ttt
 Field T defined as $ T = T^{\theta\theta} + T^{\varphi\varphi} $.
 
Scalarp_aaa
 Field A defined from X and $\mu$ insensitive to the longitudinal part of the Sym_tensor (only for $\ell \geq 2$).
 
Scalarp_tilde_b
 Field $ \tilde{B}$ defined from $ h^{rr}, \eta, W$ and h insensitive to the longitudinal part of the Sym_tensor.
 
Scalarp_tilde_c
 Field $ \tilde{C}$ defined from $ h^{rr}, \eta, W$ and h insensitive to the longitudinal part of the Sym_tensor.
 
int id_sym1
 Number of the first symmetric index (0<= id_sym1 < valence )
 
int id_sym2
 Number of the second symmetric index (id_sym1 < id_sym2 < valence )
 
const Map *const mp
 Mapping on which the numerical values at the grid points are defined.
 
int valence
 Valence of the tensor (0 = scalar, 1 = vector, etc...)
 
const Base_vecttriad
 Vectorial basis (triad) with respect to which the tensor components are defined.
 
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.
 
int n_comp
 Number of stored components, depending on the symmetry.
 
Scalar ** cmp
 Array of size n_comp of pointers onto the components.
 
const Metricmet_depend [N_MET_MAX]
 Array on the Metric 's which were used to compute derived quantities, like p_derive_cov , etc... The i-th element of this array is the Metric used to compute the i-th element of p_derive_cov , etc.
 
Tensorp_derive_cov [N_MET_MAX]
 Array of pointers on the covariant derivatives of this with respect to various metrics.
 
Tensorp_derive_con [N_MET_MAX]
 Array of pointers on the contravariant derivatives of this with respect to various metrics.
 
Tensorp_divergence [N_MET_MAX]
 Array of pointers on the divergence of this with respect to various metrics.
 

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
mapthe mapping
triad_ivectorial basis (triad) with respect to which the tensor components are defined
metthe 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
mapthe mapping
triad_ivectorial 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.
metthe metric with respect to which the divergence is defined
fichfile 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_ylma 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::Tensor::mp, Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_aaa, Lorene::Scalar::set_spectral_va(), 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

◆ 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_ylma 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(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Tensor::operator()(), and Lorene::Sym_tensor::ttt().

◆ compute_tilde_C()

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

◆ 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::dsdr(), Lorene::Tensor::mp, Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_eta, p_khi, 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::Sym_tensor::eta(), Lorene::Sym_tensor::exponential_filter_r(), 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::Sym_tensor::eta(), Lorene::Sym_tensor::exponential_filter_ylm(), 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::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Base_val::mult_x(), Lorene::Tensor::operator()(), 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::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
gammetric with respect to the transverse decomposition is performed
parparameters for the vector Poisson equation
method_poissontype 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::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::Tensor::mp, Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_mu, 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_guessa 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::maxabs(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Tensor::operator()(), 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::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Sym_tensor::mu(), Lorene::Tensor::operator()(), Lorene::Tensor_sym::position(), 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_inthe A potential (see Sym_tensor::p_aaa )
tb_inthe $\tilde{B}$ 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_inthe A potential (see Sym_tensor::p_aaa )
tb_inthe $\tilde{B}$ potential (see Sym_tensor::p_tilde_b )
tracethe 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_inthe A potential (see Sym_tensor::p_aaa )
tbtt_inthe TT-part of $\tilde{B}$ potential (see Sym_tensor::p_tilde_b and Sym_tensor::compute_tilde_B_tt() )
h_preva pointer on a guess for the trace of *this; if null, then the iteration starts from 0.
precisrelative difference in the trace computation to end the iteration.
it_maxmaximal number of iterations.

Definition at line 137 of file sym_tensor_trans_aux.C.

References Lorene::abs(), Lorene::Sym_tensor::get_tilde_B_from_TT_trace(), Lorene::Tensor::inc_dzpuis(), Lorene::max(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Sym_tensor::p_aaa, Lorene::Sym_tensor::p_tilde_b, Lorene::Sym_tensor_trans::p_trace, Lorene::Sym_tensor_trans::p_tt, 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_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::Sym_tensor::del_deriv(), 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(), and Lorene::Tensor::triad.

◆ 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
hrrthe rr component of the tensor,
mu_inthe $\mu$ potential,
precisrelative difference in the trace computation to end the iteration.
it_maxmaximal number of iterations.

Definition at line 117 of file sym_tensor_trans_aux.C.

References Lorene::Tensor::dec_dzpuis(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Sym_tensor::p_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::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::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_place_met(), 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::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::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
hijttthe TT part for this.
h_preva pointer on a guess for the trace of *this; if null, then the iteration starts from 0.
precisrelative difference in the trace computation to end the iteration.
it_maxmaximal number of iterations.

Definition at line 226 of file sym_tensor_trans_aux.C.

References Lorene::abs(), Lorene::Sym_tensor::get_tilde_B_from_TT_trace(), Lorene::Tensor::inc_dzpuis(), Lorene::max(), Lorene::Tensor::mp, Lorene::Sym_tensor_trans::p_trace, Lorene::Sym_tensor_trans::p_tt, Lorene::Sym_tensor::set_auxiliary(), Lorene::Sym_tensor_trans::sol_Dirac_tilde_B(), and Lorene::Tensor::triad.

◆ 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::Tensor::dec_dzpuis(), Lorene::Sym_tensor_trans::del_deriv(), Lorene::Tensor_sym::derive_con(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Sym_tensor_trans::p_trace, and Lorene::Sym_tensor_trans::p_tt.

◆ 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_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Map::get_mg(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Base_val::mult_x(), and R_CHEBP.

◆ 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_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Map::get_mg(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, and Lorene::Base_val::mult_x().

◆ 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_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Map::get_mg(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, and Lorene::Base_val::mult_x().

◆ 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

◆ 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

◆ 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::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Map::get_mg(), Lorene::Base_val::give_lmax(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Base_val::mult_x(), and R_CHEBP.

◆ sol_Dirac_l01_2()

void Lorene::Sym_tensor_trans::sol_Dirac_l01_2 ( const Scalar hh,
Scalar hrr,
Scalar tilde_eta,
Param par_mat 
)
inherited

Definition at line 1559 of file sym_tensor_trans_dirac_bound2.C.

◆ sol_Dirac_l01_bound()

void Lorene::Sym_tensor_trans::sol_Dirac_l01_bound ( const Scalar hh,
Scalar hrr,
Scalar tilde_eta,
Scalar bound_hrr,
Scalar bound_eta,
Param par_mat 
)
inherited

Definition at line 1583 of file sym_tensor_trans_dirac_boundfree.C.

◆ 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::Mtbl_cf::annule_hard(), Lorene::Scalar::annule_hard(), Lorene::Valeur::c, Lorene::Valeur::c_cf, Lorene::Scalar::check_dzpuis(), Lorene::Scalar::dsdr(), Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Map::get_mg(), Lorene::Base_val::give_lmax(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Base_val::mult_x(), R_CHEBP, Lorene::Valeur::set_etat_cf_qcq(), Lorene::Scalar::set_spectral_base(), Lorene::Scalar::set_spectral_va(), Lorene::Sym_tensor_trans::sol_Dirac_l01(), and Lorene::Valeur::ylm_i().

◆ 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
httthe transverse traceless part; all components must have dzpuis = 2.
precisrelative difference in the trace computation to end the iteration.
it_maxmaximal number of iterations.

Definition at line 315 of file sym_tensor_trans.C.

References Lorene::abs(), Lorene::max(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, 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
gammetric with respect to the transverse decomposition is performed
parparameters for the vector Poisson equation
method_poissontype 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::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::Tensor_sym::derive_con(), Lorene::Sym_tensor_trans::met_div, Lorene::Tensor::mp, Lorene::Sym_tensor_trans::p_tt, 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
dzpdzpuis parameter of the result, i.e. of the components $ h^{ij} $.

Definition at line 338 of file sym_tensor_tt_etamu.C.

References Lorene::Tensor::cmp, Lorene::Scalar::dec_dzpuis(), Lorene::Sym_tensor_trans::del_deriv(), Lorene::Tensor::mp, Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_eta, Lorene::Sym_tensor::p_mu, Lorene::Tensor_sym::position(), Lorene::Scalar::srdsdt(), Lorene::Scalar::srstdsdp(), and Lorene::Tensor::triad.

◆ 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::dsdt(), Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_www, 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::dsdt(), Lorene::Tensor::operator()(), Lorene::Sym_tensor::p_xxx, 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: