# [LORENE] Index convention in covariant derivatives

Eric Gourgoulhon eric.gourgoulhon at obspm.fr
Mon Dec 29 11:41:23 CET 2003

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/