Connection_flat Class Reference
[Tensorial fields]

Class Connection_flat. More...

#include <connection.h>

Inheritance diagram for Connection_flat:
Connection Connection_fcart Connection_fspher

List of all members.

Public Member Functions

 Connection_flat (const Connection_flat &)
 Copy constructor.
virtual ~Connection_flat ()
 destructor
void operator= (const Connection_flat &)
 Assignment to another Connection_flat.
virtual Tensor * p_derive_cov (const Tensor &tens) const =0
 Computes the covariant derivative $\nabla T$ of a tensor $T$ (with respect to the current connection).
virtual Tensor * p_divergence (const Tensor &tens) const =0
 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

 Connection_flat (const Map &, const Base_vect &)
 Contructor from a triad, has to be defined in the derived classes.
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_flat.

()

Abstract class for connections associated with a flat metric.

Definition at line 350 of file connection.h.

Constructor & Destructor Documentation

Connection_flat::Connection_flat ( const Map &  mpi,
const Base_vect &  bi 
) [protected]

Contructor from a triad, has to be defined in the derived classes.

Definition at line 73 of file connection_flat.C.

References Connection::assoc_metric, Connection::delta, and Tensor::set_etat_zero().

Connection_flat::Connection_flat ( const Connection_flat &  ci  ) 

Copy constructor.

Definition at line 83 of file connection_flat.C.

Connection_flat::~Connection_flat (  )  [virtual]

destructor

Definition at line 93 of file connection_flat.C.

Member Function Documentation

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

Deletes all the derived quantities.

Definition at line 201 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_flat::operator= ( const Connection_flat &   ) 

Assignment to another Connection_flat.

Reimplemented from Connection.

Reimplemented in Connection_fspher, and Connection_fcart.

Definition at line 103 of file connection_flat.C.

virtual Tensor* Connection_flat::p_derive_cov ( const Tensor &  tens  )  const [pure 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.

Reimplemented from Connection.

Implemented in Connection_fspher, and Connection_fcart.

virtual Tensor* Connection_flat::p_divergence ( const Tensor &  tens  )  const [pure 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.

Reimplemented from Connection.

Implemented in Connection_fspher, and Connection_fcart.

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

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

Reimplemented from Connection.

Definition at line 117 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 209 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 251 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 231 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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