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* Definition of Lorene class Connection
*
*/
/*
* Copyright (c) 2003-2004 Eric Gourgoulhon & Jerome Novak
*
* This file is part of LORENE.
*
* LORENE is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License version 2
* as published by the Free Software Foundation.
*
* LORENE is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with LORENE; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*
*/
#ifndef __CONNECTION_H_
#define __CONNECTION_H_
/*
* $Id: connection.h,v 1.14 2014/10/13 08:52:33 j_novak Exp $
* $Log: connection.h,v $
* Revision 1.14 2014/10/13 08:52:33 j_novak
* Lorene classes and functions now belong to the namespace Lorene.
*
* Revision 1.13 2004/03/22 13:12:40 j_novak
* Modification of comments to use doxygen instead of doc++
*
* Revision 1.12 2004/01/04 20:50:24 e_gourgoulhon
* Class Connection: data member delta is now of type Tensor_sym (and no
* longer of type Tensor_delta).
*
* Revision 1.11 2003/12/30 22:56:40 e_gourgoulhon
* Replaced member flat_conn (flat connection) by flat_met (flat metric)
* Added argument flat_met to the constructors of Connection.
* Suppressed method fait_ricci() (the computation of the Ricci is
* now devoted to the virtual method ricci()).
*
* Revision 1.10 2003/12/27 14:56:20 e_gourgoulhon
* -- Method derive_cov() suppressed.
* -- Change of the position of the derivation index from the first one
* to the last one in methods p_derive_cov() and p_divergence().
*
* Revision 1.9 2003/10/16 14:21:33 j_novak
* The calculation of the divergence of a Tensor is now possible.
*
* Revision 1.8 2003/10/06 13:58:45 j_novak
* The memory management has been improved.
* Implementation of the covariant derivative with respect to the exact Tensor
* type.
*
* Revision 1.7 2003/10/06 06:52:26 e_gourgoulhon
* Corrected documentation.
*
* Revision 1.6 2003/10/05 21:04:25 e_gourgoulhon
* Improved comments
*
* Revision 1.5 2003/10/03 14:07:23 e_gourgoulhon
* Added derived class Connection_fcart.
*
* Revision 1.4 2003/10/02 21:31:11 e_gourgoulhon
* Added methods fait_delta and update
* flat_conn is now a modifiable pointer.
*
* Revision 1.3 2003/10/02 15:44:23 j_novak
* The destructor is now public...
*
* Revision 1.2 2003/10/01 15:41:49 e_gourgoulhon
* Added mapping
*
* Revision 1.1 2003/09/29 21:14:10 e_gourgoulhon
* First version --- not ready yet.
*
*
*
* $Header: /cvsroot/Lorene/C++/Include/connection.h,v 1.14 2014/10/13 08:52:33 j_novak Exp $
*
*/
// Lorene headers
#include "tensor.h"
namespace Lorene {
class Metric ;
class Metric_flat ;
//--------------------------//
// class Connection //
//--------------------------//
/**
* Class Connection. \ingroup (tensor)
*
* 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 \f$(e_i)\f$:
* \f[
* \Gamma^i_{\ jk} := \langle e^i, \nabla_{e_k} \, e_j \rangle
* \f]
*
*/
class Connection {
// Data :
// -----
protected:
const Map* const mp ; ///< Reference mapping
/** Triad \f$(e_i)\f$ with respect to which the connection coefficients
* are defined.
*/
const Base_vect* const triad ;
/** Tensor \f$\Delta^i_{\ jk}\f$ which defines
* the connection with respect to the flat one: \f$\Delta^i_{\ jk}\f$
* is the difference between the connection coefficients
* \f$\Gamma^i_{\ jk}\f$ and
* the connection coefficients \f${\bar \Gamma}^i_{\ jk}\f$ of the
* flat connection. The connection coefficients with respect to the
* triad \f$(e_i)\f$ are defined
* according to the MTW convention:
* \f[
* \Gamma^i_{\ jk} := \langle e^i, \nabla_{e_k} \, e_j \rangle
* \f]
* Note that \f$\Delta^i_{\ jk}\f$ is symmetric with respect to the
* indices j and k.
*
*/
Tensor_sym delta ;
/** Indicates whether the connection is associated with a metric
* (in which case the Ricci tensor is symmetric, i.e. the
* actual type of \c p_ricci is a \c Sym_tensor )
*/
bool assoc_metric ;
private:
/** Flat metric with respect to which \f$\Delta^i_{\ jk}\f$
* (member \c delta ) is defined.
*
*/
const Metric_flat* flat_met ;
// Derived data :
// ------------
protected:
/// Pointer of the Ricci tensor associated with the connection
mutable Tensor* p_ricci ;
// Constructors - Destructor
// -------------------------
public:
/** Standard constructor ab initio.
*
* @param delta_i tensor \f$\Delta^i_{\ jk}\f$ which defines
* the connection with respect to the flat one: \f$\Delta^i_{\ jk}\f$
* is the difference between the connection coefficients
* \f$\Gamma^i_{\ jk}\f$ and
* the connection coefficients \f${\bar \Gamma}^i_{\ jk}\f$ of the
* flat connection. The connection coefficients with respect to the
* triad \f$(e_i)\f$ are defined according to the MTW convention:
* \f[
* \Gamma^i_{\ jk} := \langle e^i, \nabla_{e_k} \, e_j \rangle
* \f]
* \f$\Delta^i_{\ jk}\f$ must be symmetric with respect to the
* indices j and k.
* @param flat_met_i flat metric with respect to which \f$\Delta^i_{\ jk}\f$
* is defined
*
*/
Connection(const Tensor_sym& delta_i, const Metric_flat& flat_met_i) ;
/** Standard constructor for a connection associated with a metric.
*
* @param met Metric to which the connection will be associated
* @param flat_met_i flat metric to define the \f$\Delta^i_{\ jk}\f$
* representation of the connection
*
*/
Connection(const Metric& met, const Metric_flat& flat_met_i) ;
Connection(const Connection& ) ; ///< Copy constructor
protected:
/// Constructor for derived classes
Connection(const Map&, const Base_vect& ) ;
public:
virtual ~Connection() ; ///< Destructor
// Memory management
// -----------------
protected:
/// Deletes all the derived quantities
void del_deriv() const ;
/// Sets to \c 0x0 all the pointers on derived quantities
void set_der_0x0() const ;
// Mutators / assignment
// ---------------------
public:
/// Assignment to another \c Connection
void operator=(const Connection&) ;
/** Update the connection when it is defined ab initio.
*
* @param delta_i tensor \f$\Delta^i_{\ jk}\f$ which defines
* the connection with respect to the flat one: \f$\Delta^i_{\ jk}\f$
* is the difference between the connection coefficients
* \f$\Gamma^i_{\ jk}\f$ and
* the connection coefficients \f${\bar \Gamma}^i_{\ jk}\f$ of the
* flat connection.
* \f$\Delta^i_{\ jk}\f$ must be symmetric with respect to the
* indices j and k.
*/
void update(const Tensor_sym& delta_i) ;
/** Update the connection when it is associated with a metric.
*
* @param met Metric to which the connection is associated
*
*/
void update(const Metric& met) ;
// Accessors
// ---------
public:
/// Returns the mapping
const Map& get_mp() const {return *mp; } ;
/** Returns the tensor \f$\Delta^i_{\ jk}\f$ which defines
* the connection with respect to the flat one: \f$\Delta^i_{\ jk}\f$
* is the difference between the connection coefficients
* \f$\Gamma^i_{\ jk}\f$ and
* the connection coefficients \f${\bar \Gamma}^i_{\ jk}\f$ of the
* flat connection. The connection coefficients with respect to the
* triad \f$(e_i)\f$ are defined according to the MTW convention:
* \f[
* \Gamma^i_{\ jk} := \langle e^i, \nabla_{e_k} \, e_j \rangle
* \f]
* Note that \f$\Delta^i_{\ jk}\f$ is symmetric with respect to the
* indices j and k.
*
* @return \c delta}(i,j,k) = \f$\Delta^i_{\ jk}\f$
*/
const Tensor_sym& get_delta() const {return delta; } ;
// Computational methods
// ---------------------
public:
/** Computes the covariant derivative \f$\nabla T\f$ of a tensor \f$T\f$
* (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.
*/
virtual Tensor* p_derive_cov(const Tensor& tens) const ;
/** Computes the divergence of a tensor \f$T\f$
* (with respect to the current connection).
* The divergence is taken with respect of the last index of \f$T\f$
* which thus must be contravariant.
* For instance if \f$T\f$ is a twice contravariant tensor, whose
* components w.r.t. the
* triad \f$e_i\f$ are \f$T^{ij}\f$: \f$T = T^{ij} \; e_i \otimes e_j\f$,
* the divergence of \f$T\f$ is the vector
* \f[
* {\rm div}\, T = \nabla_k T^{ik} \; e_i
* \f]
* where \f$\nabla\f$ denotes the current connection.
* @param tens tensor \f$T\f$
* @return pointer on the divergence of \f$T\f$ ;
* this pointer is
* polymorphe, i.e. its is a pointer on a \c Scalar
* if \f$T\f$ is a \c Vector , on a \c Vector if \f$T\f$ is a tensor
* of valence 2, and on a \c Tensor otherwise.
* NB: The corresponding memory is allocated by the method
* \c p_divergence() and
* must be deallocated by the user afterwards.
*/
virtual Tensor* p_divergence(const Tensor& tens) const ;
/** Computes (if not up to date) and returns the Ricci tensor
* associated with the current connection
*/
virtual const Tensor& ricci() const ;
private:
/** Computes the difference \f$\Delta^i_{\ jk}\f$ between the
* connection coefficients and that a the flat connection
* in the case where the current connection is associated
* with a metric
*/
void fait_delta(const Metric& ) ;
};
//-------------------------------//
// class Connection_flat //
//-------------------------------//
/**
* Class Connection_flat. \ingroup (tensor)
*
* Abstract class for connections associated with a flat metric.
*
*/
class Connection_flat : public Connection {
// Constructors - Destructor
// -------------------------
protected:
/// Contructor from a triad, has to be defined in the derived classes
Connection_flat(const Map&, const Base_vect&) ;
public:
Connection_flat(const Connection_flat & ) ; ///< Copy constructor
virtual ~Connection_flat() ; ///< destructor
// Mutators / assignment
// ---------------------
public:
/// Assignment to another \c Connection_flat
void operator=(const Connection_flat&) ;
// Computational methods
// ---------------------
public:
/** Computes the covariant derivative \f$\nabla T\f$ of a tensor \f$T\f$
* (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.
*/
virtual Tensor* p_derive_cov(const Tensor& tens) const = 0 ;
/** Computes the divergence of a tensor \f$T\f$
* (with respect to the current connection).
* The divergence is taken with respect of the last index of \f$T\f$
* which thus must be contravariant.
* For instance if \f$T\f$ is a twice contravariant tensor, whose
* components w.r.t. the
* triad \f$e_i\f$ are \f$T^{ij}\f$: \f$T = T^{ij} \; e_i \otimes e_j\f$,
* the divergence of \f$T\f$ is the vector
* \f[
* {\rm div} T = \nabla_k T^{ik} \; e_i
* \f]
* where \f$\nabla\f$ denotes the current connection.
* @param tens tensor \f$T\f$
* @return pointer on the divergence of \f$T\f$ ;
* this pointer is
* polymorphe, i.e. its is a pointer on a \c Scalar
* if \f$T\f$ is a \c Vector , on a \c Vector if \f$T\f$ is a tensor
* of valence 2, and on a \c Tensor otherwise.
* NB: The corresponding memory is allocated by the method
* \c p_divergence() and
* must be deallocated by the user afterwards.
*/
virtual Tensor* p_divergence(const Tensor& tens) const = 0 ;
/** Computes (if not up to date) and returns the Ricci tensor
* associated with the current connection
*/
virtual const Tensor& ricci() const ;
};
//-------------------------------//
// class Connection_fspher //
//-------------------------------//
/**
* Class Connection_fspher.\ingroup (tensor)
*
* Class for connections associated with a flat metric and given onto
* an orthonormal spherical triad.
*
*/
class Connection_fspher : public Connection_flat {
// Constructors - Destructor
// -------------------------
public:
/// Contructor from a spherical flat-metric-orthonormal basis
Connection_fspher(const Map&, const Base_vect_spher&) ;
Connection_fspher(const Connection_fspher& ) ; ///< Copy constructor
public:
virtual ~Connection_fspher() ; ///<destructor
// Mutators / assignment
// ---------------------
public:
/// Assignment to another \c Connection_fspher
void operator=(const Connection_fspher&) ;
// Computational methods
// ---------------------
public:
/** Computes the covariant derivative \f$\nabla T\f$ of a tensor \f$T\f$
* (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.
*/
virtual Tensor* p_derive_cov(const Tensor& tens) const ;
/** Computes the divergence of a tensor \f$T\f$
* (with respect to the current connection).
* The divergence is taken with respect of the last index of \f$T\f$
* which thus must be contravariant.
* For instance if \f$T\f$ is a twice contravariant tensor, whose
* components w.r.t. the
* triad \f$e_i\f$ are \f$T^{ij}\f$: \f$T = T^{ij} \; e_i \otimes e_j\f$,
* the divergence of \f$T\f$ is the vector
* \f[
* {\rm div} T = \nabla_k T^{ik} \; e_i
* \f]
* where \f$\nabla\f$ denotes the current connection.
* @param tens tensor \f$T\f$
* @return pointer on the divergence of \f$T\f$ ;
* this pointer is
* polymorphe, i.e. its is a pointer on a \c Scalar
* if \f$T\f$ is a \c Vector , on a \c Vector if \f$T\f$ is a tensor
* of valence 2, and on a \c Tensor otherwise.
* NB: The corresponding memory is allocated by the method
* \c p_divergence() and
* must be deallocated by the user afterwards.
*/
virtual Tensor* p_divergence(const Tensor& tens) const ;
};
//-------------------------------//
// class Connection_fcart //
//-------------------------------//
/**
* Class Connection_fcart.\ingroup (tensor)
*
* Class for connections associated with a flat metric and given onto
* an orthonormal Cartesian triad.
*
*/
class Connection_fcart : public Connection_flat {
// Constructors - Destructor
// -------------------------
public:
/// Contructor from a Cartesian flat-metric-orthonormal basis
Connection_fcart(const Map&, const Base_vect_cart&) ;
Connection_fcart(const Connection_fcart& ) ; ///< Copy constructor
public:
virtual ~Connection_fcart() ; ///<destructor
// Mutators / assignment
// ---------------------
public:
/// Assignment to another \c Connection_fcart
void operator=(const Connection_fcart&) ;
// Computational methods
// ---------------------
public:
/** Computes the covariant derivative \f$\nabla T\f$ of a tensor \f$T\f$
* (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.
*/
virtual Tensor* p_derive_cov(const Tensor& tens) const ;
/** Computes the divergence of a tensor \f$T\f$
* (with respect to the current connection).
* The divergence is taken with respect of the last index of \f$T\f$
* which thus must be contravariant.
* For instance if \f$T\f$ is a twice contravariant tensor, whose
* components w.r.t. the
* triad \f$e_i\f$ are \f$T^{ij}\f$: \f$T = T^{ij} \; e_i \otimes e_j\f$,
* the divergence of \f$T\f$ is the vector
* \f[
* {\rm div} T = \nabla_k T^{ik} \; e_i
* \f]
* where \f$\nabla\f$ denotes the current connection.
* @param tens tensor \f$T\f$
* @return pointer on the divergence of \f$T\f$ ;
* this pointer is
* polymorphe, i.e. its is a pointer on a \c Scalar
* if \f$T\f$ is a \c Vector , on a \c Vector if \f$T\f$ is a tensor
* of valence 2, and on a \c Tensor otherwise.
* NB: The corresponding memory is allocated by the method
* \c p_divergence() and
* must be deallocated by the user afterwards.
*/
virtual Tensor* p_divergence(const Tensor& tens) const ;
};
}
#endif
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