LORENE
Lorene::Sym_tensor Class Reference

Class intended to describe valence-2 symmetric tensors. More...

#include <sym_tensor.h>

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

Public Member Functions

 Sym_tensor (const Map &map, const Itbl &tipe, const Base_vect &triad_i)
 Standard constructor.
 
 Sym_tensor (const Map &map, int tipe, const Base_vect &triad_i)
 Standard constructor when both indices are of the same type.
 
 Sym_tensor (const Sym_tensor &a)
 Copy constructor.
 
 Sym_tensor (const Tensor &a)
 Constructor from a Tensor .
 
 Sym_tensor (const Map &map, const Base_vect &triad_i, FILE *fich)
 Constructor from a file (see sauve(FILE*) ).
 
virtual ~Sym_tensor ()
 Destructor.
 
virtual void operator= (const Sym_tensor &a)
 Assignment to another 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_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).
 
virtual const Scalareta (Param *par=0x0) const
 Gives the field $\eta$ (see member p_eta ).
 
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.
 
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

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.
 

Friends

class Metric
 

Detailed Description

Class intended to describe valence-2 symmetric tensors.

The storage and the calculations are different and quicker than with an usual Tensor .

The valence must be 2.()

Definition at line 223 of file sym_tensor.h.

Constructor & Destructor Documentation

◆ Sym_tensor() [1/5]

Lorene::Sym_tensor::Sym_tensor ( const Map map,
const Itbl tipe,
const Base_vect triad_i 
)

Standard constructor.

Parameters
mapthe mapping
tipe1-D array of integers (class Itbl ) of size 2 containing the type of each index, COV for a covariant one and CON for a contravariant one, with the following storage convention:
  • tipe(0) : type of the first index
  • tipe(1) : type of the second index
triad_ivectorial basis (triad) with respect to which the tensor components are defined

Definition at line 142 of file sym_tensor.C.

References set_der_0x0().

◆ Sym_tensor() [2/5]

Lorene::Sym_tensor::Sym_tensor ( const Map map,
int  tipe,
const Base_vect triad_i 
)

Standard constructor when both indices are of the same type.

Parameters
mapthe mapping
tipethe type of the indices.
triad_ivectorial basis (triad) with respect to which the tensor components are defined

Definition at line 152 of file sym_tensor.C.

References set_der_0x0().

◆ Sym_tensor() [3/5]

Lorene::Sym_tensor::Sym_tensor ( const Sym_tensor a)

◆ Sym_tensor() [4/5]

Lorene::Sym_tensor::Sym_tensor ( const Tensor a)

Constructor from a Tensor .

The symmetry of the input tensor is assumed but is not checked.

Definition at line 193 of file sym_tensor.C.

References Lorene::Tensor::cmp, Lorene::Tensor_sym::indices(), Lorene::Tensor::n_comp, Lorene::Evolution< TyT >::position(), and set_der_0x0().

◆ Sym_tensor() [5/5]

Lorene::Sym_tensor::Sym_tensor ( const Map map,
const Base_vect triad_i,
FILE fich 
)

Constructor from a file (see 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.
fichfile which has been used by the function sauve(FILE*) .

Definition at line 210 of file sym_tensor.C.

References Lorene::Tensor::n_comp, set_der_0x0(), and Lorene::Tensor::valence.

◆ ~Sym_tensor()

Lorene::Sym_tensor::~Sym_tensor ( )
virtual

Destructor.

Definition at line 222 of file sym_tensor.C.

References del_deriv().

Member Function Documentation

◆ compute_A()

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

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()(), p_aaa, Lorene::Scalar::set_spectral_va(), Lorene::Tensor::triad, 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

◆ compute_tilde_B_tt()

Scalar Lorene::Sym_tensor::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 .

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 compute_tilde_B(), Lorene::Base_val::give_quant_numbers(), Lorene::Tensor::mp, Lorene::Tensor::operator()(), and ttt().

◆ compute_tilde_C()

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

◆ del_deriv()

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

Deletes the derived quantities.

Reimplemented from Lorene::Tensor.

Reimplemented in Lorene::Sym_tensor_trans, and Lorene::Sym_tensor_tt.

Definition at line 286 of file sym_tensor.C.

References Lorene::Tensor::del_deriv(), del_derive_met(), p_aaa, p_eta, p_mu, p_tilde_b, p_tilde_c, p_ttt, p_www, p_xxx, and set_der_0x0().

◆ del_derive_met()

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

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

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

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
virtual

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::Tensor::mp, Lorene::Tensor::operator()(), p_eta, and Lorene::Tensor::triad.

◆ exponential_filter_r()

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

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, eta(), 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. 
)
virtual

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, eta(), exponential_filter_ylm(), Lorene::Tensor::mp, mu(), Lorene::Tensor::n_comp, Lorene::Tensor::operator()(), set_auxiliary(), Lorene::Tensor::triad, ttt(), www(), and xxx().

◆ 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
protected

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
protected

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.

◆ longit_pot()

const Vector & Lorene::Sym_tensor::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).

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(), 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

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()(), p_mu, and Lorene::Tensor::triad.

◆ operator=() [1/3]

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

◆ operator=() [2/3]

void Lorene::Sym_tensor::operator= ( const Tensor a)
virtual

Assignment to a Tensor .

The symmetry is assumed but not checked.

Reimplemented from Lorene::Tensor_sym.

Reimplemented in Lorene::Sym_tensor_trans, and Lorene::Sym_tensor_tt.

Definition at line 275 of file sym_tensor.C.

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

◆ operator=() [3/3]

void Lorene::Sym_tensor::operator= ( const Tensor_sym a)
virtual

Assignment to a Tensor_sym.

Reimplemented from Lorene::Tensor_sym.

Reimplemented in Lorene::Sym_tensor_trans, and Lorene::Sym_tensor_tt.

Definition at line 267 of file sym_tensor.C.

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

◆ 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 
)

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 del_deriv(), Lorene::Scalar::mult_r_dzpuis(), p_eta, p_mu, p_ttt, p_www, p_xxx, Lorene::Tensor::set(), and Lorene::Tensor::triad.

◆ set_der_0x0()

void Lorene::Sym_tensor::set_der_0x0 ( ) const
protected

Sets the pointers on derived quantities to 0x0.

Definition at line 305 of file sym_tensor.C.

References p_aaa, p_eta, p_mu, p_tilde_b, p_tilde_c, p_ttt, p_www, p_xxx, and set_der_met_0x0().

◆ set_der_met_0x0()

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

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_longit_trans()

void Lorene::Sym_tensor::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.

(see the documentation of these derived members for details)

Definition at line 88 of file sym_tensor_decomp.C.

References Lorene::Tensor::dec_dzpuis(), del_deriv(), Lorene::Tensor::get_place_met(), p_longit_pot, p_transverse, and Lorene::Tensor::set_dependance().

◆ transverse()

const Sym_tensor_trans & Lorene::Sym_tensor::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.

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

◆ ttt()

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

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

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()(), p_www, Lorene::Scalar::stdsdp(), and Lorene::Tensor::triad.

◆ xxx()

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

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()(), p_xxx, and Lorene::Tensor::triad.

Friends And Related Symbol Documentation

◆ Metric

Definition at line 588 of file sym_tensor.h.

Member Data Documentation

◆ p_aaa

Scalar* Lorene::Sym_tensor::p_aaa
mutableprotected

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
mutableprotected

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]
mutableprotected

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
mutableprotected

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
mutableprotected

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
mutableprotected

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_transverse

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

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_ttt

Scalar* Lorene::Sym_tensor::p_ttt
mutableprotected

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
mutableprotected

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
mutableprotected

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: