Last update: 01.04.2005, 10:00 CEST

                                01.04.2004: Remark inserted in Sect.2

Remarks on the "Evans Lemma"

Gerhard W. Bruhn, Darmstadt University of Technology

1. Evans Lemma

Evans starts the "proof" of his "Evans Lemma" by applying a "covariant differentiation" operator D^μ to his so-called "tetrad postulate", the compatibility relation of frames (see e.g. [2; Sect.4])
(1.1)                                                 ∂_μq_λ^a + ω^a_μb q_λ^b − Γ^ν_μλ q_ν^a = 0 .
without telling the reader how the operator D^μ should be defined.

D^μ (∂_μq_λ^a + ω^a_μb q_λ^b − Γ^ν_μλ q_ν^a) = 0.                                 (9.36)

Further more he assumes that the operator D^μ would anihilate the tetrad coefficients q_ν^a. Since the existence of such an operator is at least dubious, we shall give an independent derivation below (see Sect. 2).

However, before let's have a look at the "Evans Lemma" itself: After having introduced the "scalar curvatures" R₁ and R₂ by

− R₁ q_λ^a := (D^μω^a_μb) q_λ^b − (D^μ Γ^ν_μλ) q_ν^a .                                 (9.44)

and

− R₂ q_λ^a := − Γ^ν_νμ ω^μa_b q_λ^b + Γ^ν_νμ Γ^μν_λ q_ν^a ,                                 (9.48)

then the main equation, the "Evans Lemma", is

o q_λ^a = R q_λ^a ,                                                                 (9.49)

where

R = R₁ + R₂.                                                                 (9.50)

We remark that the "curvatures" R₁ and R₂depend on the tetrad coefficients, while the main equation (9.49) pretends to be a single linear equation for each single tetrad coefficients. Actually, however, equ. (9.50) represents a nonlinear system of partial differential equations for the 16 tetrad coefficients q_λ^a. Especially there is no reason to consider equ. (9.49) as an eigenvalue problem of the d’Alembertian operator o, since the "eigenvalues" R are non constant and solution dependent. Therefore all conclusions contained in the following quotation from Evans' book are senseless:

"Given the tetrad postulate, the lemma shows that scalar curvature R is always an eigenvalue of the wave equation (9.49) for all spacetimes, in other words, R is quantized. The eigenoperator is the d’Alembertian operator o, and the eigenfunction in this case is the tetrad. In Sec. 3 we will show that the eigenfunction can be any differential form, thus introducing a powerful class of wave equations to differential geometry and physics. The lemma is the subsidiary proposition leading to the Evans wave Eq. (9.2) through Eq. (9.3). The lemma is an identity of differential geometry, and so is comparable in generality and power to the well known Poincaré lemma [14]. In other words, new theorems of topology can be developed from the Evans lemma in analogy with topological theorems [2,14] from the Poincaré lemma. This can be the subject of future work in mathematics, work which may lead in turn to new findings in physics based on topology. The immediate importance of the lemma to physics is that it is the subsidiary proposition leading to the Evans wave equation, which is valid for all radiated and matter fields. Equation (9.49) can be solved for R given tetrad components, or vice versa, solved for tetrad components for a given R. The equation is non-linear in the spin and Christoffel connections, but for a given R it is a linear second order partial differential wave equation, or eigenequation. In this sense it is an equation of wave mechanics and therefore of quantum mechanics, and so unifies quantum mechanics, unified field theory and general relativity. Its power is therefore apparent and the wave equation (9.49) reduces to known equations of physics [3-7] in the appropriate limits. These include the four Newton equations, the Poisson equations of gravitation and electrostatics, the Schr¨odinger, Klein-Gordon and Dirac equations, and the equations of O(3) electrodynamics. Equation (9.49) produces the quark color triplet through a choice of eigenfunction (a three-spinor of the SU(3) representation), and so unifies the gravitational and strong fields. . . ."

2. A correct wave equation

The compatibility relation of frames

(2.1)                                                 ∂_μq_λ^a + ω^a_μb q_λ^b − Γ^ν_μλ q_ν^a = 0

yields

(2.2)                                                 ∂^μ (∂_μq_λ^a + ω^a_μb q_λ^b − Γ^ν_μλ q_ν^a) = 0 .

With o₁ := ∂^μ∂_μ we obtain

(2.3)                                 o₁q_λ^a + ∂^μ (ω^a_μb q_λ^b − Γ^ν_μλ q_ν^a) = 0 .

or

(2.4)     o₁q_λ^a + (∂^μ ω^a_μb) q_λ^b − (∂^μ Γ^ν_μλ) q_ν^a + ω^a_μb (∂^μ q_λ^b) − Γ^ν_μλ (∂^μ q_ν^a) = 0 ,

where the first derivatives of q_λ^b and q_ν^a can be removed by

(2.5)         ∂^μq_λ^b = Γ^ρμ_λ q_ρ^b − ω^bμ_c q_λ^c                 and                 ∂^μq_ν^a = Γ^ρμ_ν q_ρ^a − ω^aμ_c q_ν^c ,

to obtain

(2.6)         o₁q_λ^a + (∂^μ ω^a_μc − ω^a_μb ω^bμ_c) q_λ^c − (∂^μ Γ^ρ_μλ + Γ^ν_μλ Γ^ρμ_ν) q_ρ^a + 2 ω^a_μc Γ^νμ_λ q_ν^c = 0 .

The result is a system of linear partial differential equations for the tetrad coefficients, all of which having the principal part of the wave equation. That is the substitute for the wrong "Evans Lemma" (9.49-50) in Chap. 9.2 of Evans' book.

References

[1]         M.W. Evans: GENERALLY COVARIANT UNIFIED FIELD THEORY:
                                THE GEOMETRIZATION OF PHYSICS; Web-Preprint,
                                http://www.aias.us/book01/GCUFT-Book-10.pdf

[2]         G.W. Bruhn: Covariant Derivatives and the Tetrad Postulate:
                                http://www2.mathematik.tu-darmstadt.de/~bruhn/covar_deriv.htm