Last update: 17.03.2005, 12:30 CET

Notations added in (3.1-4), Remark added in Sect. 4.

While tensors determine their components relative to a given basis *uniquely*
the converse is not true. The reason is that one can *fix* one or several indices
the result being components of a tensor of lower order. Therefore tensor components
require *additional information* about the meaning of the indices: blocked or free.
Neglecting this ambiguity is the reason of some confounding in literature.

**Example**
S.M. Carroll [1; p.89] introduces "vielbein" coefficients *e _{μ}*

Thus, we have to distinguish several cases for *e _{ν}*

(I) the case of single coefficients: The indices

(II) the case of components of the coordinate basis vectors: Here the index a is

(III) the case of components of the orthonormal basis one-forms where the index

Additionally to the cases (II) and (III) enumerated by S.M. Carroll one could consider

(IV) the case where

Generally spoken: The meaning of given tensor components is
*context sensitive*. In order to come to a unique assignment of a tensor
we have to distinguish the different cases by additional marks at the tensor components.
For that purpose we introduce a "Deblocking Dot" (DD) "**.**"
to be placed after all deblocked indices.
(That DD can be understood as a short for "..." or for "something must be appended here")

Under that convention the notation in case (I) remains unchanged while the other cases yield
the notations:

(II)
*e _{ν}*

(III)

(IV)

The deblocked indices determine what must be appended:

Latin indices as "a" refer to the tetrad basis: An

Greek indices as "

To be correct all supplements have to be combined in the order of the appearance of the corresponding indices with the tensor product Ä between them. However, as can be seen later on, that order has no influence on the resulting covariant derivatives.

Let's continue with our example:

(I)

(II)

(III)

(IV)

Basically for the calculation of covariant derivatives is the concept of
directional derivative *D*_{h} in direction of a given vector h.
Since the directional derivative depends *linearly* on the directional vector h
we have for
h
=
*h ^{μ}*
∂

(2.1)
*D*_{h}
=
*h ^{μ}*

Therefore it's sufficient to consider the special directional derivation operators
*D*_{∂μ} .

Further important rules for directional derivation are the Leibniz rules for tensor products SÄT :

(2.2)
*D*_{h}
(SÄT)
=
(*D*_{h}S)
Ä
T
+
S
Ä
(*D*_{h}T)

and for products *f *T (* f* function, T tensor)

(2.3)
*D*_{h}
(* f *T)
=
(*D*_{h}* f *)
T
+
*f*
(*D*_{h}T) ,

where for h = ∂_{μ}

(2.4)
*D*_{∂μ}*f*
=
∂_{μ}* f* .

For evaluation of general directional derivatives we need the directional derivatives of the basis vectors and covectors, the so-called structure relations

(2.5)
*D*_{∂μ}(∂_{ν})
=
Γ^{σ}_{μν}
∂_{σ} ,

(2.6)
*D*_{∂μ}(*dx ^{ν}*)
=
−
Γ

(2.7)
*D*_{∂μ}(e_{a})
=
*ω*^{b}_{μa}
e_{b} ,

(2.8)
*D*_{∂μ}(θ^{a})
=
−
*ω*^{a}_{μb}
θ^{b} ,

Covariant derivatives of tensor components cannot be calculated without knowing the intrinsic complete tensor belonging to the given components. We shall see that by continuing the example of Section 1.

Let T be a tensor to be subject to the directional derivative
*D*_{∂μ}.
T is a linear combination of tensor products of the same index type. The combination coefficients
are certain functions.
The Leibniz rules (2.2-3) transform
*D*_{∂μ}T
into a sum of tensor products where each summand contains at most one basis factor derived by
*D*_{∂μ}.
The occuring derivatives of basis elements can be eliminated by means of the structure relations
(2.5-8). Therefore the result is again a linear combination of summands of the same index type
as before.
These resulting coefficients are called the covariant derivatives *D*_{μ}
of the coefficients of T.

We shall demonstrate this with help of our examples:

(I) The tensor belonging to
e_{ν}^{a}
is T^{(I)} = e_{ν}^{a}
itself, a (0,0) tensor, a function, since (due to our convention) all indices are blocked.
Hence we obtain due to (2.4)

(3.1)
*D*^{(I)}_{μ}
*e _{ν}*

(II) The tensor belonging to
*e _{ν}*

and from (2.4) and (2.7) we obtain

hence

(3.2)

(III) The tensor belonging to
*e _{ν .}*

and from (2.4) and (2.6) we obtain

hence

(3.3)

(IV) The tensor belonging to
*e _{ν .}*

and from (2.4) and (2.6-7) we obtain

= (∂

hence

(3.4)

*D*_{∂μ}T^{(IV)}
=
*D*_{∂μ}(*dx ^{ν}*Äe

= Γ

Hence we obtain in addition to (3.4)

(3.5)
*D*^{(IV)}_{μ}
*e _{ν}*

The representation of tensors by their components is *context sensitive* and
gives rise to confoundings and confusion. If one ignores that context then the
calculation of covariant derivatives of tensor components will lead to confusion,
because in general several different covariant derivatives exist. In the example
of the tetrad coefficients *e _{ν}*

T^{a}
=
*D*^{(II)}Ù*q*^{a} ,

i.e. in detail

T^{a}_{μν}
=
*D*^{(II)}_{μ}
*q*_{ν}^{a}
−
*D*^{(II)}_{ν}
*q*_{μ}^{a}
.

[1]
S.M. Carroll: Lecture Notes on General Relativity

http://arxiv.org/pdf/gr-qc/9712019

[2]
W. A. Rodrigues Jr. and Q. A. G. de Souza:

An Ambiguous Statement Called ‘Tetrad Postulate’
and the Correct Field

Equations Satisfied by
the Tetrad Fields. 58 pages,

http://arxiv.org/abs/math-ph/0411085

[3]
G.W. Bruhn: Covariant Differentiation and the Tetrad Postulate

http://www.mathematik.tu-darmstadt.de/~bruhn/covar_deriv.htm

[4]
G.W. Bruhn and W.A. Rodrigues: Commentary to Evans' Proof of Cartan's First
Structure Equation,

http://www.mathematik.tu-darmstadt.de/~bruhn/1stCartan_equ.html

[5]
Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick:

Analysis, Manifolds and Physics

(revised edition) North-Holland, Amsterdam (1982)

[6]
S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, vol.1,

Interscience Publishers, New York, 1963