LORENE
Lorene::Sym_tensor_trans Class Reference
Tensorial fields

Transverse symmetric tensors of rank 2. More...

#include <sym_tensor.h>

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

Public Member Functions

 Sym_tensor_trans (const Map &map, const Base_vect &triad_i, const Metric &met)
 Standard constructor.
 Sym_tensor_trans (const Sym_tensor_trans &)
 Copy constructor.
 Sym_tensor_trans (const Map &map, const Base_vect &triad_i, const Metric &met, FILE *fich)
 Constructor from a file (see Tensor::sauve(FILE*) ).
virtual ~Sym_tensor_trans ()
 Destructor.
const Metric & get_met_div () const
 Returns the metric with respect to which the divergence and the trace are defined.
virtual void operator= (const Sym_tensor_trans &a)
 Assignment to another 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_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 Scalar & the_trace () const
 Returns the trace of the tensor with respect to metric *met_div.
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.
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 Vector & divergence (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_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.
const Vector & longit_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).
virtual const Scalar & eta (Param *par=0x0) const
 Gives the field $\eta$ (see member p_eta ).
const Scalar & mu (Param *par=0x0) const
 Gives the field $\mu$ (see member p_mu ).
const Scalar & www () const
 Gives the field W (see member p_www ).
const Scalar & xxx () const
 Gives the field X (see member p_xxx ).
const Scalar & ttt () const
 Gives the field T (see member p_ttt ).
const Scalar & compute_A (bool output_ylm=true, Param *par=0x0) const
 Gives the field A (see member p_aaa ).
const Scalar & compute_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 Scalar & compute_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_sym & derive_cov (const Metric &gam) const
 Returns the covariant derivative of this with respect to some metric $\gamma$.
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.
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.
Scalar & set (const Itbl &ind)
 Returns the value of a component (read/write version).
Scalar & set (int i1, int i2)
 Returns the value of a component for a tensor of valence 2 (read/write version).
Scalar & set (int i1, int i2, int i3)
 Returns the value of a component for a tensor of valence 3 (read/write version).
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).
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 Map & get_mp () const
 Returns the mapping.
const Base_vect * get_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.
int & set_index_type (int i)
 Sets the type of the index number i .
Itbl & set_index_type ()
 Sets the types of all the indices.
const Scalar & operator() (const Itbl &ind) const
 Returns the value of a component (read-only version).
const Scalar & operator() (int i1, int i2) const
 Returns the value of a component for a tensor of valence 2 (read-only version).
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).
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).
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 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_tensor * inverse () 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

const Metric *const met_div
 Metric with respect to which the divergence and the trace are defined.
Scalar * p_trace
 Trace with respect to the metric *met_div.
Sym_tensor_tt * p_tt
 Traceless part with respect to the metric *met_div.
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.
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.
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!):
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!):
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!):
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!):
Scalar * p_ttt
 Field T defined as $ T = T^{\theta\theta} + T^{\varphi\varphi} $.
Scalar * p_aaa
 Field A defined from X and $\mu$ insensitive to the longitudinal part of the Sym_tensor (only for $\ell \geq 2$).
Scalar * p_tilde_b
 Field $ \tilde{B}$ defined from $ h^{rr}, \eta, W$ and h insensitive to the longitudinal part of the Sym_tensor.
Scalar * p_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_vect * triad
 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 Metric * met_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.
Tensor * p_derive_cov [N_MET_MAX]
 Array of pointers on the covariant derivatives of this with respect to various metrics.
Tensor * p_derive_con [N_MET_MAX]
 Array of pointers on the contravariant derivatives of this with respect to various metrics.
Tensor * p_divergence [N_MET_MAX]
 Array of pointers on the divergence of this with respect to various metrics.

Detailed Description

Transverse symmetric tensors of rank 2.

()

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

Definition at line 608 of file sym_tensor.h.

Constructor & Destructor Documentation

◆ Sym_tensor_trans() [1/3]

Lorene::Sym_tensor_trans::Sym_tensor_trans ( 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 114 of file sym_tensor_trans.C.

References Lorene::Map(), met_div, set_der_0x0(), and Lorene::Sym_tensor::Sym_tensor().

◆ Sym_tensor_trans() [2/3]

Lorene::Sym_tensor_trans::Sym_tensor_trans ( const Sym_tensor_trans & source)

Copy constructor.

Definition at line 125 of file sym_tensor_trans.C.

References met_div, p_trace, p_tt, set_der_0x0(), Lorene::Sym_tensor::Sym_tensor(), and Sym_tensor_trans().

◆ Sym_tensor_trans() [3/3]

Lorene::Sym_tensor_trans::Sym_tensor_trans ( 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 139 of file sym_tensor_trans.C.

References Lorene::Map(), met_div, set_der_0x0(), and Lorene::Sym_tensor::Sym_tensor().

◆ ~Sym_tensor_trans()

Lorene::Sym_tensor_trans::~Sym_tensor_trans ( )
virtual

Destructor.

Definition at line 151 of file sym_tensor_trans.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::div_r_dzpuis(), Lorene::Scalar::div_tant(), Lorene::dsdr(), Lorene::Scalar::dsdt(), Lorene::Scalar::get_dzpuis(), Lorene::Tensor::mp, Lorene::Tensor::operator()(), p_aaa, Lorene::Scalar::poisson_angu(), Lorene::Scalar::stdsdp(), Lorene::Tensor::triad, and xxx().

◆ 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_ylma 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.

References Lorene::Scalar::annule_hard(), Lorene::Valeur::c_cf, Lorene::Scalar::div_r_dzpuis(), Lorene::Scalar::div_tant(), Lorene::Scalar::dsdr(), Lorene::Scalar::dsdt(), Lorene::Scalar::get_dzpuis(), Lorene::Scalar::get_etat(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Tensor::operator()(), p_tilde_b, Lorene::Scalar::poisson_angu(), Lorene::Scalar::set_spectral_va(), Lorene::Scalar::stdsdp(), Lorene::Tensor::triad, ttt(), www(), and Lorene::Valeur::ylm().

◆ 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::Scalar::annule_hard(), Lorene::Valeur::c, Lorene::Valeur::c_cf, compute_tilde_B(), Lorene::Scalar::div_r_dzpuis(), Lorene::Scalar::dsdr(), Lorene::Scalar::get_dzpuis(), Lorene::Scalar::get_etat(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Tensor::operator()(), Lorene::Mtbl_cf::set(), Lorene::Scalar::set_dzpuis(), Lorene::Scalar::set_spectral_va(), ttt(), Lorene::Valeur::ylm(), and Lorene::Valeur::ylm_i().

◆ 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_ylma 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.

References Lorene::Scalar::annule_hard(), Lorene::Valeur::c_cf, Lorene::Scalar::div_r_dzpuis(), Lorene::Scalar::div_tant(), Lorene::Scalar::dsdr(), Lorene::Scalar::dsdt(), Lorene::Scalar::get_dzpuis(), Lorene::Scalar::get_etat(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Tensor::operator()(), p_tilde_c, Lorene::Scalar::poisson_angu(), Lorene::Scalar::set_spectral_va(), Lorene::Scalar::stdsdp(), Lorene::Tensor::triad, ttt(), www(), and Lorene::Valeur::ylm().

◆ del_deriv()

void Lorene::Sym_tensor_trans::del_deriv ( ) const
protectedvirtual

Deletes the derived quantities.

Reimplemented from Lorene::Sym_tensor.

Reimplemented in Lorene::Sym_tensor_tt.

Definition at line 164 of file sym_tensor_trans.C.

References Lorene::Sym_tensor::del_deriv(), p_trace, p_tt, 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, p_longit_pot, p_transverse, and 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, Sym_tensor(), 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::eta ( Param * par = 0x0) const
virtualinherited

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

Reimplemented in Lorene::Sym_tensor_tt.

Definition at line 111 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()(), p_eta, Lorene::Scalar::poisson_angu(), Lorene::Scalar::stdsdp(), 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(), eta(), Lorene::Scalar::exponential_filter_r(), exponential_filter_r(), Lorene::Tensor::mp, mu(), Lorene::Tensor::n_comp, Lorene::Tensor::operator()(), set_auxiliary(), Lorene::Tensor::triad, ttt(), www(), and 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(), eta(), Lorene::Scalar::exponential_filter_ylm(), exponential_filter_ylm(), Lorene::Tensor::mp, mu(), Lorene::Tensor::n_comp, Lorene::Tensor::operator()(), set_auxiliary(), Lorene::Tensor::triad, ttt(), www(), and xxx().

◆ get_met_div()

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

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

Definition at line 669 of file sym_tensor.h.

References 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::annule_hard(), Lorene::Tbl::annule_hard(), Lorene::Valeur::c, Lorene::Valeur::c_cf, Lorene::Scalar::div_r_dzpuis(), Lorene::Scalar::dsdr(), Lorene::Scalar::get_dzpuis(), Lorene::Scalar::get_etat(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Base_val::mult_x(), Lorene::Tensor::operator()(), Lorene::Mtbl_cf::set(), Lorene::Scalar::set_dzpuis(), Lorene::Scalar::set_spectral_base(), Lorene::Scalar::set_spectral_va(), Lorene::Tensor::triad, and Lorene::Valeur::ylm().

◆ 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(), Sym_tensor(), Lorene::Tensor::triad, and Lorene::Tensor::type_indice.

◆ 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::diffrel(), divergence(), Lorene::Tensor::divergence(), Lorene::Vector::divergence(), Lorene::Tensor::get_place_met(), Lorene::maxabs(), Lorene::Tensor::mp, 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()(), p_mu, Lorene::Scalar::poisson_angu(), Lorene::Scalar::stdsdp(), and Lorene::Tensor::triad.

◆ operator=() [1/4]

void Lorene::Sym_tensor_trans::operator= ( const Sym_tensor & a)
virtual

Assignment to a Sym_tensor.

Reimplemented from Lorene::Sym_tensor.

Reimplemented in Lorene::Sym_tensor_tt.

Definition at line 202 of file sym_tensor_trans.C.

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

◆ operator=() [2/4]

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

Assignment to another Sym_tensor_trans.

Reimplemented in Lorene::Sym_tensor_tt.

Definition at line 186 of file sym_tensor_trans.C.

References del_deriv(), met_div, Lorene::Sym_tensor::operator=(), p_trace, p_tt, and Sym_tensor_trans().

◆ operator=() [3/4]

void Lorene::Sym_tensor_trans::operator= ( const Tensor & a)
virtual

Assignment to a Tensor.

Reimplemented from Lorene::Sym_tensor.

Reimplemented in Lorene::Sym_tensor_tt.

Definition at line 225 of file sym_tensor_trans.C.

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

◆ operator=() [4/4]

void Lorene::Sym_tensor_trans::operator= ( const Tensor_sym & a)
virtual

Assignment to a Tensor_sym.

Reimplemented from Lorene::Sym_tensor.

Reimplemented in Lorene::Sym_tensor_tt.

Definition at line 213 of file sym_tensor_trans.C.

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

◆ poisson()

Sym_tensor_trans Lorene::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:

\‍[ \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::Scalar::allocate_all(), Lorene::Tensor::change_triad(), Lorene::Sym_tensor::compute_A(), Lorene::Sym_tensor::compute_tilde_B_tt(), Lorene::Tensor::dec_dzpuis(), Lorene::Sym_tensor::divergence(), Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_dzpuis(), Lorene::Scalar::get_etat(), Lorene::Mg3d::get_non_axi(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_base(), Lorene::Mg3d::get_type_r(), Lorene::maxabs(), met_div, Lorene::Tensor::mp, Lorene::Tensor::operator()(), poisson(), Lorene::Tensor::set(), set_AtBtt_det_one(), Lorene::Scalar::set_dzpuis(), Lorene::Scalar::set_etat_one(), Lorene::Scalar::set_etat_zero(), Lorene::Scalar::set_grid_point(), Lorene::Scalar::set_spectral_base(), Lorene::Tensor::std_spectral_base(), Lorene::Sym_tensor::Sym_tensor(), Sym_tensor_trans(), Lorene::Tensor::triad, and Lorene::Scalar::val_grid_point().

◆ 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 )

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(), sol_Dirac_A(), 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 )

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::Scalar::check_dzpuis(), Lorene::Scalar::get_spectral_base(), Lorene::Sym_tensor::get_tilde_B_from_TT_trace(), Lorene::max(), met_div, Lorene::Tensor::mp, Lorene::Sym_tensor::p_aaa, Lorene::Sym_tensor::p_tilde_b, p_trace, p_tt, Lorene::Sym_tensor::set_auxiliary(), Lorene::Scalar::set_etat_zero(), Lorene::Scalar::set_spectral_base(), sol_Dirac_A(), sol_Dirac_tilde_B(), Sym_tensor_trans(), 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::Scalar::check_dzpuis(), del_deriv(), Lorene::Scalar::dsdt(), Lorene::Scalar::get_dzpuis(), Lorene::Scalar::lapang(), p_eta, p_mu, p_ttt, p_www, 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_trans::set_der_0x0 ( ) const
protected

Sets the pointers on derived quantities to 0x0.

Definition at line 174 of file sym_tensor_trans.C.

References p_trace, and p_tt.

◆ 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 p_longit_pot, and 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 )

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

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

◆ 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 )

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::Scalar::div_r(), Lorene::Sym_tensor_tt::eta(), Lorene::Scalar::get_spectral_base(), Lorene::Sym_tensor::get_tilde_B_from_TT_trace(), Lorene::max(), Lorene::Tensor::mp, Lorene::Sym_tensor::mu(), p_trace, p_tt, Lorene::Sym_tensor::set_auxiliary(), Lorene::Scalar::set_etat_zero(), Lorene::Scalar::set_spectral_base(), sol_Dirac_tilde_B(), Sym_tensor_trans(), 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 )

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::dec_dzpuis(), Lorene::Tensor::dec_dzpuis(), del_deriv(), Lorene::Scalar::derive_con(), get_met_div(), met_div, Lorene::Tensor::mp, p_trace, p_tt, Lorene::Scalar::poisson(), and Lorene::Sym_tensor::Sym_tensor().

◆ 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
protected

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::Tensor::annule_domain(), Lorene::Matrice::annule_hard(), Lorene::Mtbl_cf::annule_hard(), Lorene::Scalar::annule_hard(), Lorene::Tbl::annule_hard(), Lorene::Valeur::c, Lorene::Valeur::c_cf, Lorene::Mtbl_cf::dsdx(), Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Param::get_int(), Lorene::Diff_dsdx::get_matrice(), Lorene::Diff_id::get_matrice(), Lorene::Diff_sx::get_matrice(), Lorene::Diff_xdsdx::get_matrice(), Lorene::Param::get_n_int(), Lorene::Param::get_n_tbl_mod(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Param::get_tbl_mod(), Lorene::Mg3d::get_type_r(), Lorene::Base_val::give_quant_numbers(), Lorene::Matrice::inverse(), Lorene::Tensor::mp, Lorene::Scalar::mult_r(), Lorene::Base_val::mult_x(), R_CHEBP, Lorene::Matrice::set(), Lorene::Mtbl_cf::set(), Lorene::Tbl::set(), Lorene::Valeur::set_etat_cf_qcq(), Lorene::Matrice::set_etat_qcq(), Lorene::Tbl::set_etat_qcq(), Lorene::Matrice::set_lu(), Lorene::Scalar::set_spectral_base(), Lorene::Scalar::set_spectral_va(), Lorene::Mtbl_cf::val_in_bound_jk(), Lorene::Mtbl_cf::val_out_bound_jk(), Lorene::Valeur::ylm(), and Lorene::Valeur::ylm_i().

◆ 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 )

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::Tensor::annule_domain(), Lorene::Matrice::annule_hard(), Lorene::Mtbl_cf::annule_hard(), Lorene::Scalar::annule_hard(), Lorene::Tbl::annule_hard(), Lorene::Valeur::c, Lorene::Valeur::c_cf, Lorene::Mtbl_cf::dsdx(), Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Param::get_int(), Lorene::Diff_dsdx::get_matrice(), Lorene::Diff_id::get_matrice(), Lorene::Diff_sx::get_matrice(), Lorene::Diff_xdsdx::get_matrice(), Lorene::Param::get_n_int(), Lorene::Param::get_n_tbl_mod(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Param::get_tbl_mod(), Lorene::Mg3d::get_type_r(), Lorene::Base_val::give_quant_numbers(), Lorene::Matrice::inverse(), Lorene::Tensor::mp, Lorene::Scalar::mult_r(), Lorene::Base_val::mult_x(), Lorene::Matrice::set(), Lorene::Mtbl_cf::set(), Lorene::Tbl::set(), Lorene::Valeur::set_etat_cf_qcq(), Lorene::Matrice::set_etat_qcq(), Lorene::Tbl::set_etat_qcq(), Lorene::Tbl::set_etat_zero(), Lorene::Matrice::set_lu(), Lorene::Scalar::set_spectral_base(), Lorene::Scalar::set_spectral_va(), Lorene::Mtbl_cf::val_in_bound_jk(), Lorene::Mtbl_cf::val_out_bound_jk(), Lorene::Valeur::ylm(), and Lorene::Valeur::ylm_i().

◆ 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 )

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::Tensor::annule_domain(), Lorene::Matrice::annule_hard(), Lorene::Mtbl_cf::annule_hard(), Lorene::Scalar::annule_hard(), Lorene::Tbl::annule_hard(), Lorene::Valeur::c, Lorene::Valeur::c_cf, Lorene::Mtbl_cf::dsdx(), Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Param::get_int(), Lorene::Diff_dsdx::get_matrice(), Lorene::Diff_id::get_matrice(), Lorene::Diff_sx::get_matrice(), Lorene::Diff_xdsdx::get_matrice(), Lorene::Param::get_n_int(), Lorene::Param::get_n_tbl_mod(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Param::get_tbl_mod(), Lorene::Mg3d::get_type_r(), Lorene::Base_val::give_quant_numbers(), Lorene::Matrice::inverse(), Lorene::Tensor::mp, Lorene::Scalar::mult_r(), Lorene::Base_val::mult_x(), Lorene::Matrice::set(), Lorene::Mtbl_cf::set(), Lorene::Tbl::set(), Lorene::Valeur::set_etat_cf_qcq(), Lorene::Matrice::set_etat_qcq(), Lorene::Tbl::set_etat_qcq(), Lorene::Tbl::set_etat_zero(), Lorene::Matrice::set_lu(), Lorene::Scalar::set_spectral_base(), Lorene::Scalar::set_spectral_va(), Lorene::Mtbl_cf::val_in_bound_jk(), Lorene::Mtbl_cf::val_out_bound_jk(), Lorene::Valeur::ylm(), and Lorene::Valeur::ylm_i().

◆ 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 )

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::Param::add_int_mod(), Lorene::Param::add_itbl_mod(), Lorene::Param::add_matrice_mod(), Lorene::Param::add_tbl_mod(), Lorene::Tensor::annule_domain(), Lorene::Itbl::annule_hard(), Lorene::Matrice::annule_hard(), Lorene::Mtbl_cf::annule_hard(), Lorene::Scalar::annule_hard(), Lorene::Tbl::annule_hard(), Lorene::Scalar::annule_l(), Lorene::Valeur::c, Lorene::Valeur::c_cf, Lorene::Scalar::check_dzpuis(), Lorene::Param::clean_all(), Lorene::Valeur::coef_i(), Lorene::Scalar::div_r_dzpuis(), Lorene::Scalar::dsdr(), Lorene::Mtbl_cf::dsdx(), Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Param::get_int(), Lorene::Param::get_int_mod(), Lorene::Param::get_itbl_mod(), Lorene::Diff_dsdx::get_matrice(), Lorene::Diff_id::get_matrice(), Lorene::Diff_sx::get_matrice(), Lorene::Diff_xdsdx::get_matrice(), Lorene::Param::get_matrice_mod(), Lorene::Param::get_n_int(), Lorene::Param::get_n_int_mod(), Lorene::Param::get_n_itbl_mod(), Lorene::Param::get_n_matrice_mod(), Lorene::Param::get_n_tbl_mod(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Param::get_tbl_mod(), Lorene::Mg3d::get_type_p(), Lorene::Mg3d::get_type_r(), Lorene::Mg3d::get_type_t(), Lorene::Base_val::give_lmax(), Lorene::Base_val::give_quant_numbers(), Lorene::Matrice::inverse(), Lorene::Tensor::mp, Lorene::Scalar::mult_r(), Lorene::Scalar::mult_r_dzpuis(), R_CHEBP, Lorene::Itbl::set(), Lorene::Matrice::set(), Lorene::Mtbl_cf::set(), Lorene::Tbl::set(), Lorene::Valeur::set_etat_cf_qcq(), Lorene::Itbl::set_etat_qcq(), Lorene::Matrice::set_etat_qcq(), Lorene::Scalar::set_etat_qcq(), Lorene::Tbl::set_etat_qcq(), Lorene::Matrice::set_lu(), Lorene::Scalar::set_spectral_base(), Lorene::Scalar::set_spectral_va(), sol_Dirac_l01(), Lorene::Scalar::std_spectral_base(), Lorene::Mtbl_cf::val_in_bound_jk(), Lorene::Mtbl_cf::val_out_bound_jk(), Lorene::Valeur::ylm(), and Lorene::Valeur::ylm_i().

◆ 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 )

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::Param::add_int_mod(), Lorene::Param::add_itbl_mod(), Lorene::Param::add_matrice_mod(), Lorene::Param::add_tbl_mod(), Lorene::Tensor::annule_domain(), Lorene::Itbl::annule_hard(), Lorene::Matrice::annule_hard(), Lorene::Mtbl_cf::annule_hard(), Lorene::Scalar::annule_hard(), Lorene::Tbl::annule_hard(), Lorene::Scalar::annule_l(), Lorene::Valeur::c, Lorene::Valeur::c_cf, Lorene::Scalar::check_dzpuis(), Lorene::Param::clean_all(), Lorene::Scalar::div_r_dzpuis(), Lorene::Scalar::dsdr(), Lorene::Mtbl_cf::dsdx(), Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Param::get_int(), Lorene::Param::get_int_mod(), Lorene::Param::get_itbl_mod(), Lorene::Diff_dsdx::get_matrice(), Lorene::Diff_id::get_matrice(), Lorene::Diff_sx::get_matrice(), Lorene::Diff_xdsdx::get_matrice(), Lorene::Param::get_matrice_mod(), Lorene::Param::get_n_int(), Lorene::Param::get_n_int_mod(), Lorene::Param::get_n_itbl_mod(), Lorene::Param::get_n_matrice_mod(), Lorene::Param::get_n_tbl_mod(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Param::get_tbl_mod(), Lorene::Mg3d::get_type_p(), Lorene::Mg3d::get_type_r(), Lorene::Mg3d::get_type_t(), Lorene::Base_val::give_lmax(), Lorene::Base_val::give_quant_numbers(), Lorene::Matrice::inverse(), Lorene::Tensor::mp, Lorene::Scalar::mult_r(), Lorene::Scalar::mult_r_dzpuis(), R_CHEBP, Lorene::Itbl::set(), Lorene::Matrice::set(), Lorene::Mtbl_cf::set(), Lorene::Tbl::set(), Lorene::Valeur::set_etat_cf_qcq(), Lorene::Itbl::set_etat_qcq(), Lorene::Matrice::set_etat_qcq(), Lorene::Scalar::set_etat_qcq(), Lorene::Tbl::set_etat_qcq(), Lorene::Matrice::set_lu(), Lorene::Scalar::set_spectral_base(), Lorene::Scalar::set_spectral_va(), Lorene::Mtbl_cf::val_in_bound_jk(), Lorene::Mtbl_cf::val_out_bound_jk(), Lorene::Valeur::ylm(), and Lorene::Valeur::ylm_i().

◆ sol_Dirac_l01()

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

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::Param::add_matrice_mod(), Lorene::Itbl::annule_hard(), Lorene::Matrice::annule_hard(), Lorene::Mtbl_cf::annule_hard(), Lorene::Scalar::annule_hard(), Lorene::Tbl::annule_hard(), Lorene::Valeur::c_cf, Lorene::Scalar::div_r_dzpuis(), Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Diff_dsdx::get_matrice(), Lorene::Diff_id::get_matrice(), Lorene::Diff_sx::get_matrice(), Lorene::Diff_xdsdx::get_matrice(), Lorene::Param::get_matrice_mod(), Lorene::Param::get_n_matrice_mod(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Mg3d::get_type_r(), Lorene::Base_val::give_lmax(), Lorene::Base_val::give_quant_numbers(), Lorene::Matrice::inverse(), Lorene::Tensor::mp, Lorene::Base_val::mult_x(), R_CHEBP, Lorene::Itbl::set(), Lorene::Matrice::set(), Lorene::Mtbl_cf::set(), Lorene::Tbl::set(), Lorene::Matrice::set_etat_qcq(), Lorene::Tbl::set_etat_qcq(), Lorene::Matrice::set_lu(), Lorene::Scalar::set_spectral_va(), Lorene::Mtbl_cf::val_in_bound_jk(), Lorene::Mtbl_cf::val_out_bound_jk(), and Lorene::Valeur::ylm().

◆ sol_Dirac_l01_2()

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

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 )

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
protected

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::Param::add_int_mod(), Lorene::Param::add_itbl_mod(), Lorene::Param::add_matrice_mod(), Lorene::Param::add_tbl_mod(), Lorene::Tensor::annule_domain(), Lorene::Itbl::annule_hard(), Lorene::Matrice::annule_hard(), Lorene::Mtbl_cf::annule_hard(), Lorene::Scalar::annule_hard(), Lorene::Tbl::annule_hard(), Lorene::Scalar::annule_l(), Lorene::Valeur::c, Lorene::Valeur::c_cf, Lorene::Scalar::check_dzpuis(), Lorene::Param::clean_all(), Lorene::Scalar::div_r_dzpuis(), Lorene::Scalar::dsdr(), Lorene::Mtbl_cf::dsdx(), Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Param::get_int(), Lorene::Param::get_int_mod(), Lorene::Param::get_itbl_mod(), Lorene::Diff_dsdx::get_matrice(), Lorene::Diff_id::get_matrice(), Lorene::Diff_sx::get_matrice(), Lorene::Diff_xdsdx::get_matrice(), Lorene::Param::get_matrice_mod(), Lorene::Param::get_n_int(), Lorene::Param::get_n_int_mod(), Lorene::Param::get_n_itbl_mod(), Lorene::Param::get_n_matrice_mod(), Lorene::Param::get_n_tbl_mod(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_base(), Lorene::Scalar::get_spectral_va(), Lorene::Param::get_tbl_mod(), Lorene::Mg3d::get_type_p(), Lorene::Mg3d::get_type_r(), Lorene::Mg3d::get_type_t(), Lorene::Base_val::give_lmax(), Lorene::Base_val::give_quant_numbers(), Lorene::Matrice::inverse(), Lorene::Tensor::mp, Lorene::Scalar::mult_r(), Lorene::Scalar::mult_r_dzpuis(), Lorene::Base_val::mult_x(), R_CHEBP, Lorene::Itbl::set(), Lorene::Matrice::set(), Lorene::Mtbl_cf::set(), Lorene::Tbl::set(), Lorene::Valeur::set_etat_cf_qcq(), Lorene::Itbl::set_etat_qcq(), Lorene::Matrice::set_etat_qcq(), Lorene::Scalar::set_etat_qcq(), Lorene::Tbl::set_etat_qcq(), Lorene::Matrice::set_lu(), Lorene::Scalar::set_spectral_base(), Lorene::Scalar::set_spectral_va(), sol_Dirac_l01(), Lorene::Mtbl_cf::val_in_bound_jk(), Lorene::Mtbl_cf::val_out_bound_jk(), Lorene::Valeur::ylm(), 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 )

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

References Lorene::Sym_tensor::Sym_tensor(), Sym_tensor_trans(), and Lorene::Tensor::trace().

◆ the_trace()

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

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

Definition at line 270 of file sym_tensor_trans.C.

References met_div, 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 )

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

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

◆ tt_part()

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

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::dec_dzpuis(), Lorene::Scalar::derive_con(), Lorene::Scalar::get_dzpuis(), Lorene::Tensor::inc_dzpuis(), met_div, Lorene::Tensor::mp, p_tt, Lorene::Scalar::poisson(), Lorene::Sym_tensor::Sym_tensor(), 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 p_ttt, 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::div_tant(), Lorene::Scalar::dsdt(), Lorene::Tensor::operator()(), 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()(), 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
protected

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_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
mutableprotected

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
mutableprotected

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: