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


In Sect.1 we give a sketch of the basics of spacetime manifolds. Namely the tetrad coefficients qaμ 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.


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 TP at the points P of M are spanned by the tangential basis vectors eμ = 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 : TP × TPR . Therefore we can define the matrix (gμν) by

(1.1)                                 gμν := g(eμ,eν) ,

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

A linear transform L: TPTP that fulfils g(Lea,Leb = g(ea,eb) is called a Lorentz-transform. The Lorentz-transforms of TP constitute the well-known Lorentz group. All Lorentz-transforms have the property g(LV,LW) = g(V,W) for arbitrary vectors V, W in TP.

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

(1.2)                                 eμ = qμa ea .

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

(1.3)                                 gμν = g(eμ,eν) , = qμaqνb g(ea,eb) , = qμaqνb ηab .

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

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

(1.4)                                 DμF :=μF

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

(1.5)                                 DμFν :=μFν + Γμνρ Fρ

and for a (0,1)-tensor Fν we have

(1.6)                                 DμFν :=μFν − Γμρν Fρ .

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

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

(1.7)                                 DμFa :=μFa + ωμab Fb


(1.8)                                 DμFa :=μFa − ωμba Fb .

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 gμν = T μν

by qνb ηab to obtain

(2.2)                                 Raμ − ½ R qaμ = Taμ

and suppresses the tetrad index a:

                                        Rμ − ½ R qμ = Tμ                                                                         (3.18)

He now "wedges" that by qbν to obtain

(2.4)                                 Raμ Ù qbν − ½ R qaμ Ù qbν = Taμ Ù qbν

and suppresses the tetrad indices a,b

                                        Rμ Ù qν − ½ R qμ Ù qν = Tμ Ù qν .                                                 (3.25)

The wedge product used by Evans here is the wedge product of vectors A = Aμeμ:
                                AÙB = ½ (AμBνAνBμ) eμÙeν
written in short hand as
                                (AÙB)μν := AμÙBν := ½ (AμBνAνBμ) .

Evans remarks the term RμÙqν being antisymmetric like the electromagnetic field tensor Gμν . Hence he feels encouraged to try the following ansatz

                                        Gμν = G(0) (Rμν(A) − ½ R qμν(A))                                                 (3.29)


                                        Rμν(A) = Rμ Ù qν ,                 qμν(A) = qμ Ù qν .                         (3.26-27)

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

(2.5)                                 Gμν = G(0) (RaμÙ qbν − ½ R qaμ Ù qbν) .

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 χab, i.e. to insert a factor χab on the right hand side of (3.29), i.e. (2.5) in our detailed representation. Our first choice for χab is the Minkowskian ηab. However, then the right hand side of (3.29) vanishes since we have

(2.6)                                 qaμ Ù qbν ηab = qaμ qbν ηabqaν qbμ ηab = gμνgνμ = 0
(2.7)                                 Raμ Ù qbν ηab = Raμ qbν ηabRaν qbμ ηab = RμνRνμ = 0
due to the symmetry of the metric tensor gμν and of the Ricci tensor Rμν [4; (3.91)].

One could try to find a matrix (χab) 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)                                 Lca χcd Ldb = χab                 where Lea =: Lab eb.

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

Therefore we may conclude that only a trivial zero em-field Gμν 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/ui, and the scale factors are:

                                        hi = | r/ui | .                                                                         (3.8)

The unit vectors are

                                        ei = 1/hi r/ui                                                                        (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 = qij(S)duiduj ,                                                       (3.12)

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

                                        ω2 = dA = − ½ qij(A) dui Ù duj .                                           (3.13)

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

                                        ω1 = ds² = qμν(S) duμduν,                                                     (3.14)

                                        ω2 = * ω1 = dA = − ½ qμν(A) duμÙduν.                               (3.15)         WRONG!

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

The symmetric metric tensor is:

                                        h0h0  h0h1  h0h2  h0h3 
                                        h1h0  h1h1  h1h2  h1h3 
                    qμν(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 
                    qμν(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]


                                        Rμ = α qμ                                                                 (3.2)

Evans claims proportionality between the tensors Raμ and qaμ:

(4.1)                                                 Raμ = α qaμ .

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 = Rμνgμν = (Raμ ηab qbν) gμν = Raμ qμa = α qaμ qμa = 4 α ,

but the equation Raμ qμa = α qaμ qμa 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.


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

; Web-Preprint,

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