#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 $\nabla T$ 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 $\Delta^i_{\ jk}$ 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.

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 $\Delta^i_{\ jk}$ 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_flat.

()

Abstract class for connections associated with a flat metric.

Definition at line 354 of file connection.h.

Constructor & Destructor Documentation

◆ Connection_flat() [1/2]

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

protected

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

Definition at line 77 of file connection_flat.C.

References Lorene::Connection::assoc_metric, Lorene::Connection::Connection(), Lorene::Connection::delta, and Lorene::Map().

◆ Connection_flat() [2/2]

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

Copy constructor.

Definition at line 87 of file connection_flat.C.

References Lorene::Connection::Connection(), and Connection_flat().

◆ ~Connection_flat()

Lorene::Connection_flat::~Connection_flat ( )

virtual

destructor

Definition at line 97 of file connection_flat.C.

Member Function Documentation

◆ del_deriv()

void Lorene::Connection::del_deriv ( ) const

protectedinherited

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 )

privateinherited

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

inlineinherited

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

inlineinherited

Returns the mapping.

Definition at line 253 of file connection.h.

References Lorene::Map(), and mp.

◆ operator=()

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

Assignment to another Connection_flat.

Definition at line 107 of file connection_flat.C.

References Connection_flat().

◆ p_derive_cov()

virtual Tensor * Lorene::Connection_flat::p_derive_cov ( const Tensor & tens ) const

pure 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 from Lorene::Connection.

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

◆ p_divergence()

virtual Tensor * Lorene::Connection_flat::p_divergence ( const Tensor & tens ) const

pure 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 from Lorene::Connection.

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

◆ ricci()

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 121 of file connection_flat.C.

References Lorene::Connection::mp, Lorene::Connection::p_ricci, and Lorene::Connection::triad.

◆ set_der_0x0()

void Lorene::Connection::set_der_0x0 ( ) const

protectedinherited

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 )

inherited

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 )

inherited

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

protectedinherited

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

protectedinherited

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

privateinherited

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

protectedinherited

Reference mapping.

Definition at line 119 of file connection.h.

◆ p_ricci

Tensor* Lorene::Connection::p_ricci

mutableprotectedinherited

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

protectedinherited

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_flat() [1/2]

◆ Connection_flat() [2/2]

◆ ~Connection_flat()

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