Last update: 17.03.2005, 12:30 CET

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

Covariant Derivatives of Tensor Components

by Gerhard W. Bruhn and Waldyr A. Rodrigues Jr.

1. Tensors and their Components

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^a with the following words:

The vielbeins e_μ^a thus serve double duty as the components of the coordinate basis vectors in terms of the orthonormal basis vectors, and as components of the orthonormal basis one-forms in terms of the coordinate basis one-forms; while the inverse vielbeins . . .

Thus, we have to distinguish several cases for e_ν^a :

(I)   the case of single coefficients: The indices ν and a are fixed ,
(II)  the case of components of the coordinate basis vectors: Here the index a is deblocked, and
(III) the case of components of the orthonormal basis one-forms where the index ν is deblocked.

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

(IV) the case where both indices ν and a are deblocked.

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^{a .}        for "a is deblocked",
(III)                                e_{ν .}^a        for "ν is deblocked",
(IV)                                e_{ν .}^{a .}      for "ν and a are deblocked".

The deblocked indices determine what must be appended:

Latin indices as "a" refer to the tetrad basis: An upper latin index requires a supplement of the basis vectors {e_a, . . . } with index of the same name, by analogy a lower index refers to a supplementation from the cotetrad basis {θ^a}.

Greek indices as "α" refer to the coordinate basis: An upper Greek index requires a supplement of the basis vectors {∂_α, . . . } with index of the same name, by analogy a lower index refers to a supplementation from the covector basis {dx^α, . . . }.

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)   e_ν^a yields the same (0,0) tensor since containing no DD's.

(II) e_ν^{a .} yields the (1,0) tensor (vector) e_ν^a e_a where the DD was removed since being dispensable.

(III) e_{ν .}^a yields the (0,1) tensor (covector) e_ν^a dx ^ν where the DD was removed since being dispensable.

(IV) e_{ν .}^a yields the (1,1) tensor e_ν^a dx ^ν Ä e_a where the DD's were removed since being dispensable.

2. Rules for Directional Derivatives

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 ∂_μ

(2.1)                                                 D_h = h^μ D_∂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_hS) Ä T + S Ä (D_hT)

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

(2.3)                                 D_h ( f T) = (D_h f ) T + f (D_hT) ,

where for h = ∂_μ

(2.4)                                 D_∂μf = ∂_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^σ_μν dx^ν ,

(2.7)                                 D_∂μ(e_a) = ω^b_μa e_b ,

(2.8)                                 D_∂μ(θ^a) = − ω^a_μb θ^b ,

3. Covariant Derivation of Tensor Components

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_∂μT 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^a = D_μ e_ν^a = D_∂μ e_ν^a = ∂_μ e_ν^a .

(II) The tensor belonging to e_ν^{a .}   is   T^(II) = e_ν^a e_a , a (1,0) tensor or vector. The Leibniz rule (2.3) yields

                                D_∂μT^(II) = (D_∂μe_ν^a) e_a + e_ν^a (D_∂μe_a) ,

and from (2.4) and (2.7) we obtain

        D_∂μT^(II) = (∂_μe_ν^a) e_a + e_ν^a (ω^b_μa e_b) = (∂_μe_ν^a + e_ν^b ω^a_μb) e_a =: (D_μe_ν^{a .}) e_a ,

hence

(3.2)                                 D^(II)_μ e_ν^a = D_μe_ν^{a .} = ∂_μe_ν^a + e_ν^b ω^a_μb .

(III) The tensor belonging to e_{ν .}^a   is   T^(III) = e_ν^a dx^ν , a (0,1) tensor or covector or 1-form. The Leibniz rule (2.3) yields

                                D_∂μT^(III) = (D_∂μe_ν^a) dx^ν + e_ν^a (D_∂μdx^ν) ,

and from (2.4) and (2.6) we obtain

                D_∂μT^(III) = (∂_μe_ν^a) dx^ν − e_σ^a (Γ^σ_μν dx^ν) = (∂_μe_ν^a − e_σ^a Γ^σ_μν) dx^ν =: (D_μe_{ν .}^a) dx^ν ,

hence

(3.3)                                 D^(III)_μ e_ν^a = D_μe_{ν .}^a = ∂_μe_ν^a − e_σ^a Γ^σ_μν

(IV) The tensor belonging to e_{ν .}^{a .}   is   T^(IV) = e_ν^a dx^νÄe_a , a (1,1) tensor. The Leibniz rules (2.2-3) yield

                D_∂μT^(IV) = (D_∂μe_ν^a) dx^νÄe_a + e_ν^a (D_∂μdx^ν) Ä e_a + e_ν^a dx^ν Ä (D_∂μe_a) ,

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

               D_∂μT^(IV) = (∂_μe_ν^a) dx^νÄe_a − e_σ^a (Γ_μ^σ_ν dx^ν)Äe_a + e_ν^a dx^νÄ (ω_μ^b_a e_b)
                            = (∂_μe_ν^a + e_ν^b ω_μ^a_b − e_σ^a Γ_μ^σ_ν) dx^νÄe_a =: (D_μe_{ν .}^{a .}) dx^νÄe_a ,

hence

(3.4)                         D^(IV)_μ e_ν^a = D_μe_{ν .}^{a .} = ∂_μe_ν^a − e_σ^a Γ_μ^σ_ν + e_ν^b ω_μ^a_b .

Corollary

D_∂μT^(IV) = D_∂μ(dx^νÄe_ν) = (D_∂μdx^ν) Ä e_ν + dx^ν Ä (D_∂μe_ν)
= Γ_μ^ν_ρdx^ρÄ e_ν − dx^ν Ä Γ_μ^ρ_ν e_ρ = 0 .

Hence we obtain in addition to (3.4)

(3.5)                         D^(IV)_μ e_ν^a = D_μe_{ν .}^{a .} = ∂_μe_ν^a − e_σ^a Γ_μ^σ_ν + e_ν^b ω_μ^a_b = 0 .

4. Conclusion and Summary

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^a four different covariant derivatives (3.1-4) exist. So, using one of them, the user must give a precise reasoning of his choice, see e.g. S.M. Carroll's remarks [1; p.89] quoted in Section 1, which give rise to the two different covariant derivatives (3.2) and (3.3). Another application of one of the different covariant derivatives occurs in the context of Cartan's first equation, which can be written in the form

T^a = D^(II)Ùq^a ,

i.e. in detail

T^a_μν = D^(II)_μ q_ν^a − D^(II)_ν q_μ^a .

References

[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://www2.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://www2.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