Remarks by Arkadiusz Jadczyk on Evans' "Covariant Derivatives"

Indeed, Evans' exterior "covariant derivative" DÙ defined in [2,(17.16)] by

(D')                 D Ù X^a_b = d Ù X^a_b + ω^a_c Ù X^c_b .

is not covariant. The simplest example is the group SU(2) with its bi-invariant metric and basis of left-invariant vector fields.

The curvature tensor can be easily computed:

R_ijkl = ½ (δ_ilδ_jk − δ_ik δ_jl)

The connection coefficients are also easily calculated:

ω_ikj = ½ ε_ikj

(cf. for instance Coquereaux and Jadczyk, "Riemannian geometry, fiber bundles, Kaluza-Klein theories and all that", World Scientific, 1988, pp. 10-13)

Therefore ω^c_b Ù R^a_c ≠ 0 and it is evidently not a horizontal form - in other words: it does not define a tensorial object or, as physicists use to say: it is not gauge-invariant. Therefore (D') does not define a "covariant" operation.

Evans asserts in (4.8) of his GCUFT book (vol.1)

D^μ Γ^ρ_μρ = ∂^μ Γ^ρ_μρ + Γ^μρ_λ Γ^λ_μρ .                                 [1,(8)] and [2,(4.8)]

To see that the "covariant derivative of the index contracted Christoffel symbol" is not covariant, it is enough to consider the simplest 1-dimensional case. Let the manifold M be the half-line M := (0,∞). We endow it with natural coordinate x in (0,∞) and the metric tensor g₁₁(x) := 1 for all x in (0,∞). This is a flat metric, and the only Christoffel symbol in one dimension, Γ¹₁₁, vanishes identically:

Γ¹₁₁ = 0, for all x in (0,∞).

In particular Evans' D¹ Γ¹₁₁ = 0 identically.

Since D¹ Γ¹₁₁ is supposedly a scalar, it should vanish identically in any other coordinate system.

Let now x' be another coordinate, taking values in R given by x'(x) := log x , the inverse transformation is x(x') = e^x'. We can easily compute the metric tensor component in the primed system:

g_1'1'(x') = [^{∂ x}/_∂x']² = e^2x' .

This gives us the Christoffel symbol in the primed coordinate system:

Γ^1'_1'1' = ½ g^{1'1' ∂g_1'1'}/_∂x' = 1 , for all x' in R.

Therefore Evans' covariant derivative is now:

D^1' Γ^1'_1'1' = ^∂Γ1'_1'1'/_∂x' + Γ^1'1'_1' Γ^1'_1'1' = g^1'1' Γ^1'_1'1' Γ^1'_1'1' = e^−2x' · 1² = e^−2x' ≠ 0

for all x'. In particular at x'=0 (x=1) we get Evan's "covariant derivative" equal 1 while, if believed to be covariant, it should have value 0.

It is indeed hard to invent something more non-covariant!

Remarkable, that Evans' assertion could pass the refereeing of a prestigious journal.

References

[1] M.W. Evans, A Generally Covariant Wave Equation for Grand Unified Field Theory,
        Foundations of Physics Letters, Vol. 16, No. 6, Dec. 2003, pp. 513-547(35), Springer

[2] M.W. Evans, GENERALLY COVARIANT UNIFIED FIELD THEORY, Arima 2006

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