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 qaμ + ωaρs qsμ − Γσρμ qaσ = 0

to define a covariant derivative Ñρ that admits interpreting of (1) as Ñρ qaμ = 0 : Carroll defines

(2)                                                 Ñρ qaμ := ∂ρ qaμ + ωaμs qsρ − Γσρμ qaσ .

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

Using the Carroll derivative we have the "tetrad postulate"

(3)                                                                 Ñρ qaμ = 0

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

Example:

(4)                                                         Ñρ qμa := ∂ρ qμa − ωsρa qμs + Γμρσ qσa .

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

(5)                 Ñρ ηab = ∂ρηab − ωsρa ηsb − ωsρb ηas = − ωsρa ηsb − ωsρb ηas = − ωb,ρa − ωa,ρb .

### Note that the Carroll derivative of the constant ηab does NOT vanish!

M.W. Evans applies the Carroll derivative in the following way: He wants to calculate Ñρ gμν by referring to the equation gμν = ηab qaμ qbν. Using the Leibniz rule Evans concludes due to (3)

(6)                                 Ñρgμν = Ñρab qaμ qbν) = (Ñρηab) qaμ qbν + 0 + 0 .

Here he erroneously assumes the derivative Ñρ of a constant term ηab 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) Ñρηab = 0, which would prove the metric compatibility from the tetrad identity.

However, the correct result is due to (4)

(7)                           Ñρgμν = Ñρab qaμ qbν) = − (ωb,ρa + ωa,ρb) qaμ qbν = − (ων,ρμ + ωμ,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)                                                 Ñρgμν = ∂ρgμν − Γλρμgλν − Γλρνgμλ

where the term ∂ρgμν = ∂ρab qaμ qbν) is to be calculated with the Leibniz rule of partials:

Ñρ gμν = ∂ρ gμν − Γλρμ gλν − Γλρν gμλ
= ηab (∂ρqaμ) qbν + ηab qaμ (∂ρqbν) − Γλρμ gλν − Γλρν gμλ
= [ηab (∂ρqaμ) qbν − Γλρμ gλν] + [ηab qaμ (∂ρqbν) − Γλρν gμλ]
= ηab (∂ρqaμ − Γλρμqaλ) qbν + ηab qaμ (∂ρqbν − Γλρν qbλ)

By using the tetrad identity (1) we obtain finally

Ñρ gμν = ηab [(∂ρqaμ − Γλρμqaλ) qbν + qaμ (∂ρqbν − Γλρν qbλ)]
= − ηabaρc qcμ qbν + ωbρc qaμ qcν)
= − ηab ωaρc (qcμ qbν + qbμ qcν)
= − ωaρc (qcμ ηab qbν + ηabqbμ qcν)
= − ωaρc (qcμ qσa gσν + qcν qσa gσμ)
= − ωaρc qσa (qcμ gσν + qcν gσμ)
= − ωσρc (qcμ gσν + qcν gσμ)
= − (ωσρμ gσν + ωσρν gσμ)
= − (ων,ρμ + ωμ,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')                                                         Dν qμa := ∂ν qμa + Γμνλ qλa

in Evans' notation with suppressed Latin indices

Dν qμ := ∂ν qμ + Γμνλ qλ                                                         (2.182)

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

Dρ gμν = Dρab qμaqνb) = ηab [(Dρqaμ) qbν + qaμ (Dρqbν)]
= ηab [(∂ρ qμa + Γμρλ qλa) qνb + qμa (∂ρ qνb + Γνρλ qλb)].

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

(1')                                                 ∂ρ qμa − ωsρa qμs + Γμρσ qσa = 0

can be used to obtain

= − ηabsρa qμs qνb + ωsρb qνs qμa) = − ηabμρa qνb + ωνρb qμa)
= − (ωμρν + ωνρμ) .

(7')                                                           Dρgμν = − (ων,ρμ + ωμ,ρν),

which agrees with (7) though different covariant differential operators Ñρ and Dρ were used.

### References

[1]    M.W. Evans, Generally Covariant Unified Field Theory, the geometrization of physics,