#include <connection.h>

Inheritance diagram for Lorene::Connection:

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 (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 $\Delta^i_{\ jk}$ 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 113 of file connection.h.

Constructor & Destructor Documentation

◆ Connection() [1/4]

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

Standard constructor 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. The connection coefficients with respect to the triad

are defined according to the MTW convention:

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

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 129 of file connection.C.

References assoc_metric, delta, flat_met, Lorene::Tensor::get_index_type(), get_mp(), Lorene::Tensor::get_valence(), mp, set_der_0x0(), Lorene::Tensor_sym::sym_index1(), Lorene::Tensor_sym::sym_index2(), and triad.

◆ Connection() [2/4]

Lorene::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 150 of file connection.C.

References assoc_metric, delta, fait_delta(), flat_met, get_mp(), mp, set_der_0x0(), and triad.

◆ Connection() [3/4]

Lorene::Connection::Connection ( const Connection & conn_i )

Copy constructor.

Definition at line 166 of file connection.C.

References assoc_metric, Connection(), delta, flat_met, mp, set_der_0x0(), and triad.

◆ Connection() [4/4]

Lorene::Connection::Connection	(	const Map &	mpi,
		const Base_vect &	bi )

protected

Constructor for derived classes.

Definition at line 179 of file connection.C.

References assoc_metric, delta, flat_met, Lorene::Map(), mp, set_der_0x0(), and triad.

◆ ~Connection()

Lorene::Connection::~Connection ( )

virtual

Destructor.

Definition at line 195 of file connection.C.

References del_deriv().

Member Function Documentation

◆ del_deriv()

void Lorene::Connection::del_deriv ( ) const

protected

Deletes all the derived quantities.

Definition at line 205 of file connection.C.

References p_ricci, and set_der_0x0().

◆ fait_delta()

void Lorene::Connection::fait_delta ( const Metric & gam )

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 278 of file connection.C.

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

◆ get_delta()

const Tensor_sym & Lorene::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 271 of file connection.h.

References delta.

◆ get_mp()

const Map & Lorene::Connection::get_mp ( ) const

inline

Returns the mapping.

Definition at line 253 of file connection.h.

References Lorene::Map(), and mp.

◆ operator=()

void Lorene::Connection::operator= ( const Connection & ci )

Assignment to another Connection.

Definition at line 225 of file connection.C.

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

◆ p_derive_cov()

Tensor * Lorene::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 \f$T\f$)
of \f$\nabla T\f$ is chosen to be the \b last  one.
This convention agrees with that of MTW (see Eq. (10.17) of MTW).
For instance, if \f$T\f$ is a 1-form, whose components
w.r.t. the triad \f$e^i\f$ are \f$T_i\f$: \f$T=T_i \; e^i\f$,
then the covariant derivative of \f$T\f$ is the bilinear form
\f$\nabla T\f$ whose components \f$\nabla_j T_i\f$ are
such that 
\f[
 \nabla T = \nabla_j T_i \; e^i \otimes e^j
\f]

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

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

Definition at line 307 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::Itbl::set(), Lorene::Tensor::set(), Lorene::Tensor_sym::sym_index1(), Lorene::Tensor_sym::sym_index2(), and triad.

◆ p_divergence()

Tensor * Lorene::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

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

Definition at line 463 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::set(), Lorene::Tensor_sym::sym_index1(), Lorene::Tensor_sym::sym_index2(), and triad.

◆ ricci()

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 662 of file connection.C.

References assoc_metric, Lorene::Scalar::dec_dzpuis(), delta, flat_met, mp, p_ricci, Lorene::Scalar::set_etat_zero(), and triad.

◆ set_der_0x0()

void Lorene::Connection::set_der_0x0 ( ) const

protected

Sets to 0x0 all the pointers on derived quantities.

Definition at line 213 of file connection.C.

References p_ricci.

◆ update() [1/2]

void Lorene::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 255 of file connection.C.

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

◆ update() [2/2]

void Lorene::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 235 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().

Member Data Documentation

◆ assoc_metric

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.

◆ delta

Tensor_sym Lorene::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 141 of file connection.h.

◆ flat_met

const Metric_flat* Lorene::Connection::flat_met

private

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

Definition at line 156 of file connection.h.

◆ mp

const Map* const Lorene::Connection::mp

protected

Reference mapping.

Definition at line 119 of file connection.h.

◆ p_ricci

Tensor* Lorene::Connection::p_ricci

mutableprotected

Pointer of the Ricci tensor associated with the connection.

Definition at line 164 of file connection.h.

◆ triad

const Base_vect* const Lorene::Connection::triad

protected

Triad $(e_i)$ 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:

Public Member Functions

Protected Member Functions

Protected Attributes

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ Connection() [1/4]

◆ Connection() [2/4]

◆ Connection() [3/4]

◆ Connection() [4/4]

◆ ~Connection()

Member Function Documentation

◆ del_deriv()

◆ fait_delta()

◆ get_delta()

◆ get_mp()

◆ operator=()

◆ p_derive_cov()

◆ p_divergence()

◆ ricci()

◆ set_der_0x0()

◆ update() [1/2]

◆ update() [2/2]

Member Data Documentation

◆ assoc_metric

◆ delta

◆ flat_met

◆ mp

◆ p_ricci

◆ triad