Section 3 appended on March 6, 2006
Section 4 appended on March 7, 2006

The Central Error of Myron W. Evans'
ECE Theory − a Type Mismatch

Gerhard W. Bruhn, Darmstadt University of Technology

Summary

In Sect.1 we give a sketch of the basics of spacetime manifolds. Namely the tetrad coefficients q_a^μ are introduced which M. W. Evans believes to be an essential tool of argumentation leading far beyond the limitations of General Relativity because of giving the opportunity of modelling several other force fields of modern physics in addition to gravitation. However, as we shall see in Sect.2, the main errors of that "theory" are invalid field definitions: They are simply invalid and therefore useless due to type mismatch. This is caused by M.W.Evans' bad habit of suppressing seemingly unimportant indices. It is shown that there is no possibility of removing that tetrad indices a,b from Evans' field theory, i.e. M.W.Evans' ECE Theory cannot be repaired. In Sect.3 M.W. Evans' concept of a non-Minkowskian spacetime manifold [1; Sect.2],[2; Chap.3.2], is shown to be erroneous. In Section 4 another erroneous claim of [1; Sect.3],[2; Chap.3.3] is discussed.

Preface

The following review of M.W. Evans' "Einstein Cartan Evans Theory" refers to Evans' book [2; Chap.3] and his preceding almost identical article [1] as well. Quotes from Evans' texts will be displayed in black while the comments appear in blue. Equation labels at the right hand side are referring to [2].

1. What M.W. Evans should have given first:
A clear description of his basic assumptions

Evans constructs his spacetime by a dubious alternative method we shall discuss in Sect.3 . Here we sketch the usual method of constructing the 4-dimensional manifold M of General Relativity, see e.g. [3].

The tangent spaces T_P at the points P of M are spanned by the tangential basis vectors e_μ = ∂_μ (μ=0,1,2,3) at the respective points P of M.

There is a pseudo-metric defined at the points P of M as a bilinear function g : T_P × T_P → R . Therefore we can define the matrix (g_μν) by

(1.1) g_μν := g(e_μ,e_ν) ,

which is assumed to be of Lorentzian signature, i.e. there exist vectors e_a (a = 0, 1, 2, 3) in each T_P such that we have g(e_a,e_b) = η_ab where the matrix (η_ab) is the diagonal matrix diag(−1, +1, +1, +1). We say also the signature of the metric (g_μν) is supposed to be Lorentzian, i.e. (− ,+,+,+).

A linear transform L: T_P → T_P that fulfils g(Le_a,Le_b = g(e_a,e_b) is called a Lorentz-transform. The Lorentz-transforms of T_P constitute the well-known Lorentz group. All Lorentz-transforms have the property g(LV,LW) = g(V,W) for arbitrary vectors V, W in T_P.

Each set of orthonormalized vectors e_a (a = 0, 1, 2, 3), in T_P is called a tetrad at the point P. We assume that a certain tetrad being chosen at each T_P of the manifold M. Then we have linear representations of the coordinate basis vectors e_μ = ∂_μ (μ=0,1,2,3) by the tetrad vectors at P:

(1.2) e_μ = q_μ^a e_a .

From (1.1) and (1.2) we obtain due to the bilinearity of g

(1.3) g_μν = g(e_μ,e_ν) , = q_μ^aq_ν^b g(e_a,e_b) , = q_μ^aq_ν^b η_ab .

Therefore the matrix (g_μν) is symmetric. And more general also g(V,W) = g(W,V) for arbitrary vectors V, W of T_P. In addition the multiplication theorem for determinants yields the matrix (g_μν) to be nonsingular.

A (non-Riemannian) linear connection is supposed, i.e. we have covariant derivatives D_μ in direction of e_μ given by

(1.4) D_μF := ∂_μF

for functions F (=(0,0)-tensors), while a (1,0)-tensor F^ν has the derivative

(1.5) D_μF^ν := ∂_μF^ν + Γ_μ^ν_ρ F^ρ

and for a (0,1)-tensor F_ν we have

(1.6) D_μF_ν := ∂_μF_ν − Γ_μ^ρ_ν F_ρ .

For coordinate dependent quantities the connection causes the additional terms in Eqns.(1.5-6) with the coefficients Γ_μ^ρ_ν

By the analogue way the connection gives rise to additional terms with coefficients ω_μ^a_b for the covariant derivatives of tetrad dependent quantities, namely

(1.7) D_μF^a := ∂_μF^a + ω_μ^a_b F^b

and

(1.8) D_μF_a := ∂_μF_a − ω_μ^b_a F_b .

2. Evans' "Generally Covariant Field Equation" [1; Sect.3], [2; Chap.3.3]

Evans "multiplies" Einstein's Field equation

(2.1) R^μν − ½ R g^μν = T^μν

by q_ν^b η_ab to obtain

(2.2) R_a^μ − ½ R q_a^μ = T_a^μ

and suppresses the tetrad index a:

R^μ − ½ R q^μ = T^μ (3.18)

He now "wedges" that by q_b^ν to obtain

(2.4) R_a^μ Ù q_b^ν − ½ R q_a^μ Ù q_b^ν = T_a^μ Ù q_b^ν

and suppresses the tetrad indices a,b

R^μ Ù q^ν − ½ R q^μ Ù q^ν = T^μ Ù q^ν . (3.25)

Remark
The wedge product used by Evans here is the wedge product of vectors A = A^μe_μ:
AÙB = ½ (A^μB^ν − A^νB^μ) e_μÙe_ν
written in short hand as
(AÙB)^μν := A^μÙB^ν := ½ (A^μB^ν − A^νB^μ) .

Evans remarks the term R^μÙq^ν being antisymmetric like the electromagnetic field tensor G^μν . Hence he feels encouraged to try the following ansatz

G^μν = G⁽⁰⁾ (R^μν(A) − ½ R q^μν(A)) (3.29)

where

R^μν(A) = R^μ Ù q^ν , q^μν(A) = q^μ Ù q^ν . (3.26-27)

Thus, Evans' ansatz (3.29) with written tetrad indices is

(2.5) G^μν = G⁽⁰⁾ (R_a^μÙ q_b^ν − ½ R q_a^μ Ù q_b^ν) .

However, by comparing the left hand side and the right hand side it is evident that the ansatz cannot be correct due to type mismatch: The tetrad indices are not available at the left hand side.

Evans' field ansatz (3.29) is unjustified due to type mismatch.

The tetrad indices a,b must be removed legally . The only way to do so is to sum over a,b with some weight factors χ^ab, i.e. to insert a factor χ^ab on the right hand side of (3.29), i.e. (2.5) in our detailed representation. Our first choice for χ^ab is the Minkowskian η^ab. However, then the right hand side of (3.29) vanishes since we have

(2.6) q_a^μ Ù q_b^ν η^ab = q_a^μ q_b^ν η^ab − q_a^ν q_b^μ η^ab = g^μν − g^νμ = 0
and
(2.7) R_a^μ Ù q_b^ν η^ab = R_a^μ q_b^ν η^ab − R_a^ν q_b^μ η^ab = R^μν − R^νμ = 0
due to the symmetry of the metric tensor g^μν and of the Ricci tensor R^μν [4; (3.91)].

One could try to find a matrix (χ^ab) different from the Minkowskian to remove the indices a,b from Evans equations (3.25-29). That matrix should not depend on the special tetrad under consideration i.e. be invariant under arbitrary Lorentz transforms L:

(2.8) L_c^a χ^cd L_d^b = χ^ab where Le_a =: L_a^b e_b.

However, due to the definition of the Lorentz transforms the matrices λ (η^ab) with some factor λ are the only matrices with that property.

Therefore we may conclude that only a trivial zero em-field G^μν can fulfil the corrected Evans field ansatz.

The correction of Evans' antisymmetric field ansatz (3.29) yields
the trivial zero case merely and is irreparably therefore.

3. Further Remarks

The following remarks concern Evans' idea of the spacetime manifold as represented in his [1; Sec.3.2].

He starts with a curvilinear parameter representation r = r(u₁, u₂, u₃) in a space the property of which is not explicitely described but turns out to be an Euclidean R³ due to the Eqns.(3.10) below.

Restrict attention initially to three non-Euclidean space dimensions. The set of curvilinear coordinates is defined as (u₁, u₂, u₃), where the functions are single valued and continuously differentiable, and where there is a one to one relation between (u₁, u₂, u₃) and the Cartesian coordinates. The position vector is r(u₁, u₂, u₃), and the arc length is the modulus of the infinitesimal displacement vector:

ds = | dr | = | ^∂r/_∂u₁ du₁ + ^∂r/_∂u₂ du₂ + ^∂r/_∂u₃ du₃ | . (3.7)

The metric coefficients are ^∂r/_{∂u_i}, and the scale factors are:

h_i = | ^∂r/_{∂u_i} | . (3.8)

The unit vectors are

e_i = ¹/_{h_i} ^∂r/_{∂u_i} (3.9)

and form the O(3) symmetry cyclic relations:

e₁ × e₂ = e₃, e₂ × e₃ = e₁, e₃ × e₁ = e₂, (3.10)

where O(3) is the rotation group of three dimensional space [3-8]. The curvilinear coordinates are orthogonal if:

e₁ · e₂ = 0, e₂ · e₃ = 0, e₃ · e₁ = 0, (3.11)

The symmetric metric tensor is then defined through the line element, a one form of differential geometry NO! A symmetric TWO-form:

ω₁ = ds² = q^ij(S)du_idu_j , (3.12)

and the anti-symmetric metric tensor through the area element, a two form of differential geometry:

ω₂ = dA = − ½ q^ij(A) du_i Ù du_j . (3.13)

These results generalize as follows to the four dimensions of any non-Euclidean space-time:

ω₁ = ds² = q^μν(S) du_μdu_ν, (3.14)

ω₂ = * ω₁ = dA = − ½ q^μν(A) du_μÙdu_ν. (3.15) WRONG!

In differential geometry the element du_σ is dual to the wedge product du_μÙdu_ν. WRONG! NOT in 4-D.

The symmetric metric tensor is:

h₀h₀ h₀h₁ h₀h₂ h₀h₃
h₁h₀ h₁h₁ h₁h₂ h₁h₃
q^μν(S) = , (3.16)
h₂h₀ h₂h₁ h₂h₂ h₂h₃
h₃h₀ h₃h₁ h₃h₂ h₃h₃

and the anti-symmetric metric tensor is:

0 −h₀h₁ −h₀h₂ −h₀h₃
h₁h₀   0     −h₁h₂ h₁h₃
q^μν(A) = . (3.17)
h₂h₀ h₂h₁ 0    −h₂h₃
h₃h₀ −h₃h₁ h₃h₂    0

The symmetric metric (3.16) cannot be correct since having a vanishing determinant: All line vectors are parallel. The reason is that the author has forgotten to insert the scalar products of his basis vectors. A similiar argument holds for Equ.(3.17) being dubious.

However, even if one avoids all possibilities mentioned above of going astray Evans' method has one crucial shortcoming: The metric definable by that method. As follows from (3.7) we have ds² > 0, i.e. the metric is positive definite. That is a heritage of Evans' construction of spacetime as an embedding into a real Euclidian space (defining the metric) that one cannot get rid off.

Evans' construction cannot yield a spacetime with
local Minkowskian i.e. indefinite metric.

That was the reason why we sketched the correct method of constructing the spacetime manifold of General Relativity at the beginning of this article in Sect.1. Evans' alternative method of [1; Chap.3.2] is useless.

4. A Remark on [1; Sect.4], [2; Chap.3.4]

With

R^μ = α q^μ (3.2)

Evans claims proportionality between the tensors R_a^μ and q_a^μ:

(4.1) R_a^μ = α q_a^μ .

However, there is no proof in [1; Sect.4]/[2; Chap.3.4] available. Indeed, if we assume (4.1) then we obtain the curvature

(4.2) R = R^μνg_μν = (R_a^μ η^ab q_b^ν) g_μν = R_a^μ q_μ^a = α q_a^μ q_μ^a = 4 α ,

but the equation R_a^μ q_μ^a = α q_a^μ q_μ^a may have other solutions than (4.1). Hence there is no way from (4.2) back to (4.1).

The considerations of [1; Chap.3.4] may be based on a logical flaw.

References

[1] M.W. Evans, A Generally Covariant Field Equation for Gravitation and Electromagnetism,
Foundations of Physics Letters Vol.16 No.4, 369-377

[2] M.W. Evans, GENERALLY COVARIANT UNIFIED FIELD THEORY:
THE GEOMETRIZATION OF PHYSICS; Web-Preprint,
http://www.atomicprecision.com/new/Evans-Book-Final.pdf

[3] S. M. Carroll, Lecture Notes in Relativity", arXiv [math-ph/0411085]

The Central Error of Myron W. Evans' ECE Theory − a Type Mismatch