Connection_fspher Class Reference
[Tensorial fields]

Class Connection_fspher. More...

#include <connection.h>

Inheritance diagram for Connection_fspher:

Connection_flat Connection List of all members.

Public Member Functions

 Connection_fspher (const Map &, const Base_vect_spher &)
 Contructor from a spherical flat-metric-orthonormal basis.
 Connection_fspher (const Connection_fspher &)
 Copy constructor.
virtual ~Connection_fspher ()
 destructor
void operator= (const Connection_fspher &)
 Assignment to another Connection_fspher.
virtual Tensor * p_derive_cov (const Tensor &tens) const
 Computes the covariant derivative $\nabla T$ of a tensor $T$ (with respect to the current connection).
virtual Tensor * p_divergence (const Tensor &tens) const
 Computes the divergence of a tensor $T$ (with respect to the current connection).
virtual const Tensor & ricci () const
 Computes (if not up to date) and returns the Ricci tensor associated with the current connection.
void update (const Tensor_sym &delta_i)
 Update the connection when it is defined ab initio.
void update (const Metric &met)
 Update the connection when it is associated with a metric.
const Map & get_mp () const
 Returns the mapping.
const Tensor_sym & get_delta () const
 Returns the tensor $\Delta^i_{\ jk}$ which defines the connection with respect to the flat one: $\Delta^i_{\ jk}$ is the difference between the connection coefficients $\Gamma^i_{\ jk}$ and the connection coefficients ${\bar \Gamma}^i_{\ jk}$ of the flat connection.

Protected Member Functions

void del_deriv () const
 Deletes all the derived quantities.
void set_der_0x0 () const
 Sets to 0x0 all the pointers on derived quantities.

Protected Attributes

const Map *const mp
 Reference mapping.
const Base_vect *const triad
 Triad $(e_i)$ with respect to which the connection coefficients are defined.
Tensor_sym delta
 Tensor $\Delta^i_{\ jk}$ which defines the connection with respect to the flat one: $\Delta^i_{\ jk}$ is the difference between the connection coefficients $\Gamma^i_{\ jk}$ and the connection coefficients ${\bar \Gamma}^i_{\ jk}$ of the flat connection.
bool assoc_metric
 Indicates whether the connection is associated with a metric (in which case the Ricci tensor is symmetric, i.e.
Tensor * p_ricci
 Pointer of the Ricci tensor associated with the connection.

Detailed Description

Class Connection_fspher.

()

Class for connections associated with a flat metric and given onto an orthonormal spherical triad.

Definition at line 448 of file connection.h.

Constructor & Destructor Documentation

Connection_fspher::Connection_fspher (  const Map &  ,
const Base_vect_spher & 
) 
 

Contructor from a spherical flat-metric-orthonormal basis.

Definition at line 135 of file connection_fspher.C.

Connection_fspher::Connection_fspher (  const Connection_fspher &   ) 
 

Copy constructor.

Definition at line 141 of file connection_fspher.C.

Connection_fspher::~Connection_fspher (   )  [virtual]
 

destructor

Definition at line 151 of file connection_fspher.C.

Member Function Documentation

void Connection::del_deriv (   )  const [protected, inherited]
 

Deletes all the derived quantities.

Definition at line 198 of file connection.C.

References Connection::p_ricci, and Connection::set_der_0x0().

const Tensor_sym& Connection::get_delta (   )  const [inline, inherited]
 

Returns the tensor $\Delta^i_{\ jk}$ which defines the connection with respect to the flat one: $\Delta^i_{\ jk}$ is the difference between the connection coefficients $\Gamma^i_{\ jk}$ and the connection coefficients ${\bar \Gamma}^i_{\ jk}$ of the flat connection.

The connection coefficients with respect to the triad $(e_i)$ are defined according to the MTW convention:

\[ \Gamma^i_{\ jk} := \langle e^i, \nabla_{e_k} \, e_j \rangle \]

Note that $\Delta^i_{\ jk}$ is symmetric with respect to the indices j and k.

Returns:
delta}(i,j,k) = $\Delta^i_{\ jk}$

Definition at line 267 of file connection.h.

References Connection::delta.

const Map& Connection::get_mp (   )  const [inline, inherited]
 

Returns the mapping.

Definition at line 249 of file connection.h.

References Connection::mp.

void Connection_fspher::operator= (  const Connection_fspher &   ) 
 

Assignment to another Connection_fspher.

Definition at line 161 of file connection_fspher.C.

Tensor * Connection_fspher::p_derive_cov (  const Tensor &  tens  )  const [virtual]
 

Computes the covariant derivative $\nabla T$ of a tensor $T$ (with respect to the current connection).

The extra index (with respect to the indices of $T$) of $\nabla T$ is chosen to be the last one. This convention agrees with that of MTW (see Eq. (10.17) of MTW). For instance, if $T$ is a 1-form, whose components w.r.t. the triad $e^i$ are $T_i$: $T=T_i \; e^i$, then the covariant derivative of $T$ is the bilinear form $\nabla T$ whose components $\nabla_j T_i$ are such that

\[ \nabla T = \nabla_j T_i \; e^i \otimes e^j \]

Parameters:
tens tensor $T$
Returns:
pointer on the covariant derivative $\nabla T$ ; this pointer is polymorphe, i.e. it is a pointer on a Vector if the argument is a Scalar , and on a Tensor otherwise. NB: The corresponding memory is allocated by the method p_derive_cov() and must be deallocated by the user afterwards.

Implements Connection_flat.

Definition at line 176 of file connection_fspher.C.

References Tensor::get_index_type(), Tensor::get_n_comp(), Tensor::get_triad(), Tensor::get_valence(), Connection::mp, Itbl::set(), Tensor_sym::sym_index1(), Tensor_sym::sym_index2(), and Connection::triad.

Tensor * Connection_fspher::p_divergence (  const Tensor &  tens  )  const [virtual]
 

Computes the divergence of a tensor $T$ (with respect to the current connection).

The divergence is taken with respect of the last index of $T$ which thus must be contravariant. For instance if $T$ is a twice contravariant tensor, whose components w.r.t. the triad $e_i$ are $T^{ij}$: $T = T^{ij} \; e_i \otimes e_j$, the divergence of $T$ is the vector

\[ {\rm div} T = \nabla_k T^{ik} \; e_i \]

where $\nabla$ denotes the current connection.

Parameters:
tens tensor $T$
Returns:
pointer on the divergence of $T$ ; this pointer is polymorphe, i.e. its is a pointer on a Scalar if $T$ is a Vector , on a Vector if $T$ is a tensor of valence 2, and on a Tensor otherwise. NB: The corresponding memory is allocated by the method p_divergence() and must be deallocated by the user afterwards.

Implements Connection_flat.

Definition at line 424 of file connection_fspher.C.

References Tensor::get_index_type(), Tensor::get_n_comp(), Tensor::get_triad(), Tensor::get_valence(), Connection::mp, Itbl::set(), Tensor_sym::sym_index1(), Tensor_sym::sym_index2(), and Connection::triad.

const Tensor & Connection_flat::ricci (   )  const [virtual, inherited]
 

Computes (if not up to date) and returns the Ricci tensor associated with the current connection.

Reimplemented from Connection.

Definition at line 114 of file connection_flat.C.

References Connection::mp, Connection::p_ricci, Tensor::set_etat_zero(), and Connection::triad.

void Connection::set_der_0x0 (   )  const [protected, inherited]
 

Sets to 0x0 all the pointers on derived quantities.

Definition at line 206 of file connection.C.

References Connection::p_ricci.

void Connection::update (  const Metric &  met  )  [inherited]
 

Update the connection when it is associated with a metric.

Parameters:
met Metric to which the connection is associated

Definition at line 248 of file connection.C.

References Connection::assoc_metric, Connection::del_deriv(), Connection::fait_delta(), and Connection::flat_met.

void Connection::update (  const Tensor_sym &  delta_i  )  [inherited]
 

Update the connection when it is defined ab initio.

Parameters:
delta_i tensor $\Delta^i_{\ jk}$ which defines the connection with respect to the flat one: $\Delta^i_{\ jk}$ is the difference between the connection coefficients $\Gamma^i_{\ jk}$ and the connection coefficients ${\bar \Gamma}^i_{\ jk}$ of the flat connection. $\Delta^i_{\ jk}$ must be symmetric with respect to the indices j and k.

Definition at line 228 of file connection.C.

References Connection::assoc_metric, Connection::del_deriv(), Connection::delta, Connection::flat_met, Tensor::get_index_type(), Tensor::get_valence(), Tensor_sym::sym_index1(), and Tensor_sym::sym_index2().

Member Data Documentation

bool Connection::assoc_metric [protected, inherited]
 

Indicates whether the connection is associated with a metric (in which case the Ricci tensor is symmetric, i.e.

the actual type of p_ricci is a Sym_tensor )

Definition at line 143 of file connection.h.

Tensor_sym Connection::delta [protected, inherited]
 

Tensor $\Delta^i_{\ jk}$ which defines the connection with respect to the flat one: $\Delta^i_{\ jk}$ is the difference between the connection coefficients $\Gamma^i_{\ jk}$ and the connection coefficients ${\bar \Gamma}^i_{\ jk}$ of the flat connection.

The connection coefficients with respect to the triad $(e_i)$ are defined according to the MTW convention:

\[ \Gamma^i_{\ jk} := \langle e^i, \nabla_{e_k} \, e_j \rangle \]

Note that $\Delta^i_{\ jk}$ is symmetric with respect to the indices j and k.

Definition at line 137 of file connection.h.

const Map* const Connection::mp [protected, inherited]
 

Reference mapping.

Definition at line 115 of file connection.h.

Tensor* Connection::p_ricci [mutable, protected, inherited]
 

Pointer of the Ricci tensor associated with the connection.

Definition at line 160 of file connection.h.

const Base_vect* const Connection::triad [protected, inherited]
 

Triad $(e_i)$ with respect to which the connection coefficients are defined.

Definition at line 120 of file connection.h.

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