[Tensorial fields]

Class Connection_flat. More...

`#include <connection.h>`

Inheritance diagram for Lorene::Connection_flat:

## 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 of a tensor (with respect to the current connection). | |

virtual Tensor * | p_divergence (const Tensor &tens) const =0 |

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. | |

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. | |

## 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 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. |

Class Connection_flat.

()

Abstract class for connections associated with a flat metric.

Definition at line 354 of file connection.h.

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

Definition at line 80 of file connection_flat.C.

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

Lorene::Connection_flat::Connection_flat | ( | const Connection_flat & | ci |
) |

Copy constructor.

Definition at line 90 of file connection_flat.C.

Lorene::Connection_flat::~Connection_flat | ( | ) | ` [virtual]` |

destructor

Definition at line 100 of file connection_flat.C.

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

Deletes all the derived quantities.

Definition at line 208 of file connection.C.

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

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

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.

**Returns:**`delta}`

(i,j,k) =

Definition at line 271 of file connection.h.

References Lorene::Connection::delta.

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

Returns the mapping.

Definition at line 253 of file connection.h.

References Lorene::Connection::mp.

void Lorene::Connection_flat::operator= | ( | const Connection_flat & | ) |

Assignment to another `Connection_flat`

.

Reimplemented from Lorene::Connection.

Reimplemented in Lorene::Connection_fspher, and Lorene::Connection_fcart.

Definition at line 110 of file connection_flat.C.

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

**Parameters:**-
*tens*tensor

**Returns:**- pointer on the covariant derivative ; 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 Lorene::Connection.

Implemented in Lorene::Connection_fspher, and Lorene::Connection_fcart.

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.

**Parameters:**-
*tens*tensor

**Returns:**- pointer on the divergence of ; this pointer is polymorphe, i.e. its is a pointer on a
`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 from Lorene::Connection.

Implemented in Lorene::Connection_fspher, and Lorene::Connection_fcart.

const Tensor & Lorene::Connection_flat::ricci | ( | ) | const` [virtual]` |

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

Reimplemented from Lorene::Connection.

Definition at line 124 of file connection_flat.C.

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

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

Sets to `0x0`

all the pointers on derived quantities.

Definition at line 216 of file connection.C.

References Lorene::Connection::p_ricci.

void Lorene::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 258 of file connection.C.

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

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

Update the connection when it is defined ab initio.

**Parameters:**-
*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 Lorene::Connection::assoc_metric, Lorene::Connection::del_deriv(), Lorene::Connection::delta, Lorene::Connection::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, 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 147 of file connection.h.

Tensor_sym Lorene::Connection::delta` [protected, inherited]` |

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 Map* const Lorene::Connection::mp` [protected, inherited]` |

Reference mapping.

Definition at line 119 of file connection.h.

Tensor* Lorene::Connection::p_ricci` [mutable, protected, inherited]` |

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, inherited]` |

Triad with respect to which the connection coefficients are defined.

Definition at line 124 of file connection.h.

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

Generated on 7 Dec 2019 for LORENE by 1.6.1