Connection Class Reference
[Tensorial fields]

Class Connection. More...

#include <connection.h>

Inheritance diagram for Connection:

Connection_flat Connection_fcart Connection_fspher List of all members.

Public Member Functions

 Connection (const Tensor_sym &delta_i, const Metric_flat &flat_met_i)
 Standard constructor ab initio.
 Connection (const Metric &met, const Metric_flat &flat_met_i)
 Standard constructor for a connection associated with a metric.
 Connection (const Connection &)
 Copy constructor.
virtual ~Connection ()
 Destructor.
void operator= (const Connection &)
 Assignment to another 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.
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.

Protected Member Functions

 Connection (const Map &, const Base_vect &)
 Constructor for 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.

Private Member Functions

void fait_delta (const Metric &)
 Computes the difference $\Delta^i_{\ jk}$ between the connection coefficients and that a the flat connection in the case where the current connection is associated with a metric.

Private Attributes

const Metric_flat * flat_met
 Flat metric with respect to which $\Delta^i_{\ jk}$ (member delta ) is defined.

Detailed Description

Class Connection.

()

This class deals only with torsion-free connections.

Note that we use the MTW convention for the indices of the connection coefficients with respect to a given triad $(e_i)$:

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

Definition at line 109 of file connection.h.

Constructor & Destructor Documentation

Connection::Connection (  const Tensor_sym &  delta_i,
const Metric_flat &  flat_met_i
) 
 

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

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

$\Delta^i_{\ jk}$ must be symmetric with respect to the indices j and k.

flat_met_i flat metric with respect to which $\Delta^i_{\ jk}$ is defined

Definition at line 122 of file connection.C.

References Tensor::get_index_type(), Tensor::get_valence(), set_der_0x0(), Tensor_sym::sym_index1(), and Tensor_sym::sym_index2().

Connection::Connection (  const Metric &  met,
const Metric_flat &  flat_met_i
) 
 

Standard constructor for a connection associated with a metric.

Parameters:
met Metric to which the connection will be associated
flat_met_i flat metric to define the $\Delta^i_{\ jk}$ representation of the connection

Definition at line 143 of file connection.C.

References fait_delta(), and set_der_0x0().

Connection::Connection (  const Connection &   ) 
 

Copy constructor.

Definition at line 159 of file connection.C.

References set_der_0x0().

Connection::Connection (  const Map &  ,
const Base_vect & 
)  [protected]
 

Constructor for derived classes.

Definition at line 172 of file connection.C.

References set_der_0x0().

Connection::~Connection (   )  [virtual]
 

Destructor.

Definition at line 188 of file connection.C.

References del_deriv().

Member Function Documentation

void Connection::del_deriv (   )  const [protected]
 

Deletes all the derived quantities.

Definition at line 198 of file connection.C.

References p_ricci, and set_der_0x0().

void Connection::fait_delta (  const Metric &   )  [private]
 

Computes the difference $\Delta^i_{\ jk}$ between the connection coefficients and that a the flat connection in the case where the current connection is associated with a metric.

Definition at line 271 of file connection.C.

References Metric::con(), Metric::cov(), delta, Tensor_sym::derive_cov(), flat_met, and Tensor::set().

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

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 delta.

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

Returns the mapping.

Definition at line 249 of file connection.h.

References mp.

void Connection::operator= (  const Connection &   ) 
 

Assignment to another Connection.

Definition at line 218 of file connection.C.

References del_deriv(), delta, flat_met, and triad.

Tensor * Connection::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.

Reimplemented in Connection_flat, Connection_fspher, and Connection_fcart.

Definition at line 300 of file connection.C.

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

Tensor * Connection::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.

Reimplemented in Connection_flat, Connection_fspher, and Connection_fcart.

Definition at line 456 of file connection.C.

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

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

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

Reimplemented in Connection_flat.

Definition at line 655 of file connection.C.

References assoc_metric, delta, Tensor_sym::derive_cov(), flat_met, mp, p_ricci, Scalar::set_etat_zero(), Tensor::set_etat_zero(), and triad.

void Connection::set_der_0x0 (   )  const [protected]
 

Sets to 0x0 all the pointers on derived quantities.

Definition at line 206 of file connection.C.

References p_ricci.

void Connection::update (  const Metric &  met  ) 
 

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 assoc_metric, del_deriv(), fait_delta(), and flat_met.

void Connection::update (  const Tensor_sym &  delta_i  ) 
 

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 assoc_metric, del_deriv(), delta, 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]
 

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]
 

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 Metric_flat* Connection::flat_met [private]
 

Flat metric with respect to which $\Delta^i_{\ jk}$ (member delta ) is defined.

Definition at line 152 of file connection.h.

const Map* const Connection::mp [protected]
 

Reference mapping.

Definition at line 115 of file connection.h.

Tensor* Connection::p_ricci [mutable, protected]
 

Pointer of the Ricci tensor associated with the connection.

Definition at line 160 of file connection.h.

const Base_vect* const Connection::triad [protected]
 

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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