#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. More...

	Connection (const Metric &met, const Metric_flat &flat_met_i)
	Standard constructor for a connection associated with a metric. More...

	Connection (const Connection &)
	Copy constructor. More...

virtual	~Connection ()
	Destructor. More...

void	operator= (const Connection &)
	Assignment to another `Connection`. More...

void	update (const Tensor_sym &delta_i)
	Update the connection when it is defined ab initio. More...

void	update (const Metric &met)
	Update the connection when it is associated with a metric. More...

const Map &	get_mp () const
	Returns the mapping. More...

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. More...

virtual Tensor *	p_derive_cov (const Tensor &tens) const
	Computes the covariant derivative $\nabla T$ of a tensor (with respect to the current connection). More...

virtual Tensor *	p_divergence (const Tensor &tens) const
	Computes the divergence of a tensor (with respect to the current connection). More...

virtual const Tensor &	ricci () const
	Computes (if not up to date) and returns the Ricci tensor associated with the current connection. More...

Protected Member Functions
	Connection (const Map &, const Base_vect &)
	Constructor for derived classes. More...

void	del_deriv () const
	Deletes all the derived quantities. More...

void	set_der_0x0 () const
	Sets to `0x0` all the pointers on derived quantities. More...

Protected Attributes
const Map *const	mp
	Reference mapping. More...

const Base_vect *const	triad
	Triad with respect to which the connection coefficients are defined. More...

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. More...

bool	assoc_metric
	Indicates whether the connection is associated with a metric (in which case the Ricci tensor is symmetric, i.e. More...

Tensor *	p_ricci
	Pointer of the Ricci tensor associated with the connection. More...

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. More...

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

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 $\Delta^i_{\ jk}$ 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 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().

◆ 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 153 of file connection.C.

References fait_delta(), and set_der_0x0().

◆ Connection() [3/4]

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

Copy constructor.

Definition at line 169 of file connection.C.

References set_der_0x0().

◆ Connection() [4/4]

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

protected

Constructor for derived classes.

Definition at line 182 of file connection.C.

References set_der_0x0().

◆ ~Connection()

Lorene::Connection::~Connection ( )

virtual

Destructor.

Definition at line 198 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 208 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 281 of file connection.C.

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

◆ 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 mp.

◆ operator=()

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

Assignment to another Connection.

Definition at line 228 of file connection.C.

References 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 $T$ ) 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

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

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::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_fspher, and Lorene::Connection_flat.

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.

◆ 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 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.

◆ set_der_0x0()

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.

◆ update() [1/2]

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 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().

◆ update() [2/2]

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 258 of file connection.C.

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

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