Class Connection. More...
#include <connection.h>
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 which defines the connection with respect to the flat one: is the difference between the connection coefficients and the connection coefficients of the flat connection. | |
virtual Tensor * | p_derive_cov (const Tensor &tens) const |
Computes the covariant derivative of a tensor (with respect to the current connection). | |
virtual Tensor * | p_divergence (const Tensor &tens) const |
Computes the divergence of a tensor (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 with respect to which the connection coefficients are defined. | |
Tensor_sym | delta |
Tensor which defines the connection with respect to the flat one: is the difference between the connection coefficients and the connection coefficients 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 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 (member delta ) is defined. |
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 :
Definition at line 113 of file connection.h.
Lorene::Connection::Connection | ( | const Tensor_sym & | delta_i, | |
const Metric_flat & | flat_met_i | |||
) |
Standard constructor ab initio.
delta_i | tensor which defines the connection with respect to the flat one: is the difference between the connection coefficients and the connection coefficients of the flat connection. The connection coefficients with respect to the triad are defined according to the MTW convention:
must be symmetric with respect to the indices j and k. | |
flat_met_i | flat metric with respect to which is defined |
Definition at line 132 of file connection.C.
References Lorene::Tensor::get_index_type(), Lorene::Tensor::get_valence(), set_der_0x0(), Lorene::Tensor_sym::sym_index1(), and Lorene::Tensor_sym::sym_index2().
Lorene::Connection::Connection | ( | const Metric & | met, | |
const Metric_flat & | flat_met_i | |||
) |
Standard constructor for a connection associated with a metric.
met | Metric to which the connection will be associated | |
flat_met_i | flat metric to define the representation of the connection |
Definition at line 153 of file connection.C.
References fait_delta(), and set_der_0x0().
Lorene::Connection::Connection | ( | const Connection & | conn_i | ) |
Constructor for derived classes.
Definition at line 182 of file connection.C.
References set_der_0x0().
Lorene::Connection::~Connection | ( | ) | [virtual] |
void Lorene::Connection::del_deriv | ( | ) | const [protected] |
Deletes all the derived quantities.
Definition at line 208 of file connection.C.
References p_ricci, and set_der_0x0().
void Lorene::Connection::fait_delta | ( | const Metric & | gam | ) | [private] |
Computes the difference 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 281 of file connection.C.
References Lorene::Metric::con(), Lorene::Metric::cov(), delta, Lorene::Tensor_sym::derive_cov(), flat_met, and Lorene::Tensor::set().
const Tensor_sym& Lorene::Connection::get_delta | ( | ) | const [inline] |
Returns the tensor which defines the connection with respect to the flat one: is the difference between the connection coefficients and the connection coefficients of the flat connection.
The connection coefficients with respect to the triad are defined according to the MTW convention:
Note that is symmetric with respect to the indices j and k.
delta}
(i,j,k) = Definition at line 271 of file connection.h.
References delta.
const Map& Lorene::Connection::get_mp | ( | ) | const [inline] |
void Lorene::Connection::operator= | ( | const Connection & | ci | ) |
Assignment to another Connection
.
Reimplemented in Lorene::Connection_flat, Lorene::Connection_fspher, and Lorene::Connection_fcart.
Definition at line 228 of file connection.C.
References del_deriv(), delta, flat_met, and triad.
Computes the covariant derivative of a tensor (with respect to the current connection).
The extra index (with respect to the indices of ) of is chosen to be the last one. This convention agrees with that of MTW (see Eq. (10.17) of MTW). For instance, if is a 1-form, whose components w.r.t. the triad are : , then the covariant derivative of is the bilinear form whose components are such that
tens | tensor |
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 Lorene::Connection_flat, Lorene::Connection_fspher, and Lorene::Connection_fcart.
Definition at line 310 of file connection.C.
References Lorene::Scalar::dec_dzpuis(), delta, Lorene::Tensor::derive_cov(), flat_met, Lorene::Tensor::get_index_type(), Lorene::Tensor::get_n_comp(), Lorene::Tensor::get_triad(), Lorene::Tensor::get_valence(), Lorene::Tensor::indices(), mp, Lorene::Tensor::set(), Lorene::Itbl::set(), Lorene::Tensor_sym::sym_index1(), Lorene::Tensor_sym::sym_index2(), and triad.
Computes the divergence of a tensor (with respect to the current connection).
The divergence is taken with respect of the last index of which thus must be contravariant. For instance if is a twice contravariant tensor, whose components w.r.t. the triad are : , the divergence of is the vector
where denotes the current connection.
tens | tensor |
Scalar
if is a Vector
, on a Vector
if 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 Lorene::Connection_flat, Lorene::Connection_fspher, and Lorene::Connection_fcart.
Definition at line 466 of file connection.C.
References Lorene::Scalar::dec_dzpuis(), delta, Lorene::Tensor::divergence(), flat_met, Lorene::Tensor::get_index_type(), Lorene::Tensor::get_n_comp(), Lorene::Tensor::get_triad(), Lorene::Tensor::get_valence(), Lorene::Tensor::indices(), mp, Lorene::Itbl::set(), Lorene::Tensor_sym::sym_index1(), Lorene::Tensor_sym::sym_index2(), Lorene::Tensor::trace(), and triad.
const Tensor & Lorene::Connection::ricci | ( | ) | const [virtual] |
Computes (if not up to date) and returns the Ricci tensor associated with the current connection.
Reimplemented in Lorene::Connection_flat.
Definition at line 665 of file connection.C.
References assoc_metric, Lorene::Scalar::dec_dzpuis(), delta, Lorene::Tensor_sym::derive_cov(), flat_met, mp, p_ricci, Lorene::Tensor::set(), Lorene::Scalar::set_etat_zero(), and triad.
void Lorene::Connection::set_der_0x0 | ( | ) | const [protected] |
Sets to 0x0
all the pointers on derived quantities.
Definition at line 216 of file connection.C.
References p_ricci.
void Lorene::Connection::update | ( | const Metric & | met | ) |
Update the connection when it is associated with a metric.
met | Metric to which the connection is associated |
Definition at line 258 of file connection.C.
References assoc_metric, del_deriv(), fait_delta(), and flat_met.
void Lorene::Connection::update | ( | const Tensor_sym & | delta_i | ) |
Update the connection when it is defined ab initio.
delta_i | tensor which defines the connection with respect to the flat one: is the difference between the connection coefficients and the connection coefficients of the flat connection. must be symmetric with respect to the indices j and k. |
Definition at line 238 of file connection.C.
References assoc_metric, del_deriv(), delta, flat_met, Lorene::Tensor::get_index_type(), Lorene::Tensor::get_valence(), Lorene::Tensor_sym::sym_index1(), and Lorene::Tensor_sym::sym_index2().
bool Lorene::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 147 of file connection.h.
Tensor_sym Lorene::Connection::delta [protected] |
Tensor which defines the connection with respect to the flat one: is the difference between the connection coefficients and the connection coefficients of the flat connection.
The connection coefficients with respect to the triad are defined according to the MTW convention:
Note that is symmetric with respect to the indices j and k.
Definition at line 141 of file connection.h.
const Metric_flat* Lorene::Connection::flat_met [private] |
Flat metric with respect to which (member delta
) is defined.
Definition at line 156 of file connection.h.
const Map* const Lorene::Connection::mp [protected] |
Reference mapping.
Definition at line 119 of file connection.h.
Tensor* Lorene::Connection::p_ricci [mutable, protected] |
Pointer of the Ricci tensor associated with the connection.
Definition at line 164 of file connection.h.
const Base_vect* const Lorene::Connection::triad [protected] |
Triad with respect to which the connection coefficients are defined.
Definition at line 124 of file connection.h.