[LORENE] Index convention in covariant derivatives

[Eric Gourgoulhon]

Dear All,

The convention about the index storage of covariant
derivatives has been changed to let it agree with
that of MTW (cf. Eq. (10.17) of MTW). For any tensor
T and any connection \nabla, the extra index
(with respect to the indices of T) of \nabla T is chosen
to be the *last* one.
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
Note the order of indices i and j: although \nabla
appears first, its index is the second in the tensor
\nabla T.
This convention follows from the ";" notation of
covariant derivatives and is well spread in the
litterature, even in books which use the notation
nabla instead of ";" (e.g. MTW).

The new convention has been implemented in methods
     Connection::p_derive_cov()
     Connection::p_divergence()
     Tensor::derive_cov()
     Tensor::derive_con()
     Tensor::divergence()

Note that it disagrees with the convention adopted
for the old class Tenseur.

Note also that, in order not to duplicate pieces of
code, the method
     Connection::derive_cov()
which returned an *object* of class Tensor, has been
suppressed, leaving only
     Connection::p_derive_cov()
which returns a (polymorph) *pointer* on a Tensor.
This last method is less user friendly (the pointer
addresses some memory space allocated by "new" which
must be de-allocated by hand ("delete")), but
in practice, most of covariant derivatives are
computed for a connection associated with some metric.
In this case, the method
     Tensor::derive_cov(const Metric& met)
is to be used instead, and is very user friendly:
it returns an always up to date reference on the
covariant derivative of the tensor w.r.t. the metric met.

Bonne annee 2004 !

	Eric.

-- 
Eric Gourgoulhon
Laboratoire de l'Univers et de ses THeories (LUTH)
UMR 8102 du CNRS / Observatoire de Paris, F-92195 Meudon Cedex, France
tel: +33 1.45.07.74.33 (secretariat : +33 1.45.07.75.10)
e-mail: Eric.Gourgoulhon at obspm.fr    WWW: http://www.luth.obspm.fr/