Last Update: Sept 18, 2005, 08:00 pm

Evans Proving the Metric Compatibility?

by Gerhard W. Bruhn, Darmstadt University of Technology

The definition of covariant derivatives is not uniquely determined. S.M. Carroll [2] used the well known tetrad identity

(1)                                                 ∂_ρ q^a_μ + ω^a_ρs q^s_μ − Γ^σ_ρμ q^a_σ = 0

to define a covariant derivative Ñ_ρ that admits interpreting of (1) as Ñ_ρ q^a_μ = 0 : Carroll defines

(2)                                                 Ñ_ρ q^a_μ := ∂_ρ q^a_μ + ω^a_μs q^s_ρ − Γ^σ_ρμ q^a_σ .

In the following we'll speak here of Ñ_ρ as of the "Carroll derivative" for matter of distinction.

Using the Carroll derivative we have the "tetrad postulate"

(3)                                                                 Ñ_ρ q^a_μ = 0

The rule: Each upper Latin index (e.g. at q^a_· ) causes an additive term + ω^a_μb q^b_· , while a lower Latin index (e.g. at q^·_a) gives rise to the additive term − ω^a_μb q^·_a. Greek indices have to be treated as usual [2; (3.1),(3.12)]

Example:

(4)                                                         Ñ_ρ q^μ_a := ∂_ρ q^μ_a − ω^s_ρa q^μ_s + Γ^μ_ρσ q^σ_a .

Let η = (η_ab) := diag(−1, 1, 1, 1) be the Minkowski matrix. Then due to the constancy of each η_ab we obtain

(5)                 Ñ_ρ η_ab = ∂_ρη_ab − ω^s_ρa η_sb − ω^s_ρb η_as = − ω^s_ρa η_sb − ω^s_ρb η_as = − ω_b,ρa − ω_a,ρb .

Note that the Carroll derivative of the constant η_ab does NOT vanish!

M.W. Evans applies the Carroll derivative in the following way: He wants to calculate Ñ_ρ g_μν by referring to the equation g_μν = η_ab q^a_μ q^b_ν. Using the Leibniz rule Evans concludes due to (3)

(6)                                 Ñ_ρg_μν = Ñ_ρ (η_ab q^a_μ q^b_ν) = (Ñ_ρη_ab) q^a_μ q^b_ν + 0 + 0 .

Here he erroneously assumes the derivative Ñ_ρ of a constant term η_ab to be ZERO (also - among other errors - e.g. on [1; p.46 and p.70] where the Minkowski matrix is unwritten to have an elegant notation) Ñ_ρη_ab = 0, which would prove the metric compatibility from the tetrad identity.

However, the correct result is due to (4)

(7)                           Ñ_ρg_μν = Ñ_ρ (η_ab q^a_μ q^b_ν) = − (ω_b,ρa + ω_a,ρb) q^a_μ q^b_ν = − (ω_ν,ρμ + ω_μ,ρν)

which differs from zero in general.

Evans should have recognized his conclusion to be wrong, since one can get the desired result directly also by evaluating :

(8)                                                 Ñ_ρg_μν = ∂_ρg_μν − Γ^λ_ρμg_λν − Γ^λ_ρνg_μλ

                Ñ_ρ g_μν = ∂_ρ g_μν − Γ^λ_ρμ g_λν − Γ^λ_ρν g_μλ
                           = η_ab (∂_ρq^a_μ) q^b_ν + η_ab q^a_μ (∂_ρq^b_ν) − Γ^λ_ρμ g_λν − Γ^λ_ρν g_μλ
                           = [η_ab (∂_ρq^a_μ) q^b_ν − Γ^λ_ρμ g_λν] + [η_ab q^a_μ (∂_ρq^b_ν) − Γ^λ_ρν g_μλ]
                           = η_ab (∂_ρq^a_μ − Γ^λ_ρμq^a_λ) q^b_ν + η_ab q^a_μ (∂_ρq^b_ν − Γ^λ_ρν q^b_λ)

By using the tetrad identity (1) we obtain finally

                Ñ_ρ g_μν = η_ab [(∂_ρq^a_μ − Γ^λ_ρμq^a_λ) q^b_ν + q^a_μ (∂_ρq^b_ν − Γ^λ_ρν q^b_λ)]
                           = − η_ab (ω^a_ρc q^c_μ q^b_ν + ω^b_ρc q^a_μ q^c_ν)
                           = − η_ab ω^a_ρc (q^c_μ q^b_ν + q^b_μ q^c_ν)
                           = − ω^a_ρc (q^c_μ η_ab q^b_ν + η_abq^b_μ q^c_ν)
                           = − ω^a_ρc (q^c_μ q^σ_a g_σν + q^c_ν q^σ_a g_σμ)
                           = − ω^a_ρc q^σ_a (q^c_μ g_σν + q^c_ν g_σμ)
                           = − ω^σ_ρc (q^c_μ g_σν + q^c_ν g_σμ)
                           = − (ω^σ_ρμ g_σν + ω^σ_ρν g_σμ)
                           = − (ω_ν,ρμ + ω_μ,ρν).

in accordance with Equ.(7). Therefore we have

The tetrad postulate (3) does not imply the metric compatibility of the connection.

Remark Evans reference [2] to the covariant derivative to be used is ambiguous. S.M. Carroll gives two different definitions: The "Carroll definition" at [2; p.91] and the "usual" one for D at [2; p.56 (3.1) and p.58 (3.12)]. At [1; p.46 (2.182) we find the application of the usual definition [2; p.56 (3.1)]

(4')                                                         D_ν q^μ_a := ∂_ν q^μ_a + Γ^μ_νλ q^λ_a

With the extended equation (4') instead of (2.182) we obtain

                                D_ρ g^μν = D_ρ (η^ab q^μ_aq^ν_b) = η^ab [(D_ρq^a_μ) q^b_ν + q^a_μ (D_ρq^b_ν)]
                                = η^ab [(∂_ρ q^μ_a + Γ^μ_ρλ q^λ_a) q^ν_b + q^μ_a (∂_ρ q^ν_b + Γ^ν_ρλ q^λ_b)].

Here the analoguous equation to the tetrade identity (1), the equation

(1')                                                 ∂_ρ q^μ_a − ω^s_ρa q^μ_s + Γ^μ_ρσ q^σ_a = 0

can be used to obtain

                                = − η^ab (ω^s_ρa q^μ_s q^ν_b + ω^s_ρb q^ν_s q^μ_a) = − η^ab (ω^μ_ρa q^ν_b + ω^ν_ρb q^μ_a)
                                = − (ω^μ_ρ^ν + ω^ν_ρ^μ) .

The reader is kindly asked to prove the analoguous equation

which agrees with (7) though different covariant differential operators Ñ_ρ and D_ρ were used.

References

[1]    M.W. Evans, Generally Covariant Unified Field Theory, the geometrization of physics,
                http://www.aias.us/Comments/Evans-Book-Final.pdf

[2]    Sean M. Carroll, Lecture Notes on General Relativity,
                http://arxiv.org/pdf/gr-qc/9712019