Last update: March 02, 2006


Sketch of Myron W. Evans' ECE Theory

Gerhard W. Bruhn, Darmstadt University of Technology


Summary

As we shall see in Sect.1 Evans introduces the tetrad coefficient qaμ, which he 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. In Sect.2 we'll see that Evans due to calculation errors has derived a wrong "Evans Wave Equation" for the tetrad coefficients. However, the most important point is, that Evans uses the tetrad coefficients for "developing" a wrong "Evans Field Theory" (Sect.3). The main errors of that "theory" are invalid field definitions: They are simply invalid and therefore useless due to type mismatch. That type mismatch is caused by the bad habit of suppressing seemingly unimportant indices. There is no possibility of removing that tetrad indices a,b from Evans' field theory, i.e. Evans' ECE Theory cannot be repaired.


Preface

The following review of M.W. Evans' "Einstein Cartan Evans Theory" refers to Evans' book [1]. Quotes from Evans text will be displayed in black while the comments appear in blue. Equation labels at the right hand side are referring to [1].


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

Evans considers the spacetime as 4-dimensional manifold M. 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 nonsingular and 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.

A (non-Riemannian) linear connection is supposed, i.e. we have covariant derivatives Dμ of the 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

and

(1.8)                                 DμFa :=μFa − ωμba Fb .


2. Differential Equations for the tetrad coefficients qμa

A well-known identity between coordinate frames and tetrads [5; (3.5)] says

(2.1)                                 ∂μqνa + ωμab qνb − Γμσν qσa = 0 ,

which is a first order linear system of pde for the tetrad coefficients.

We apply the operator ∂μ = gμττ to Equ.(2.1) to obtain with o :=μμ

(2.2)                                 oqνa + ∂μ (ωμab qνb − Γμσν qσa) = 0 .

or

(2.3)     oqνa + (∂μ ωμab) qνb − (∂μ Γμσν) qσa + ωμab (∂μ qνb) − Γμσν (∂μ qσa) = 0 ,

where the first derivatives of qνb and qσa can be removed by

(2.4)         ∂μqνb = Γμρν qρbωμbc qλc         and         ∂μqσa = Γμρσ qρaωμac qσc ,

to obtain

(2.5)   oqνa+(∂μωμacωμab ωμbc) qνc− (∂μΓμρν+ Γμσν Γμρσ) qρa+ 2ωμac Γμσν qσc = 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 Wave Equation" (4.16) together with (4.8) in [1; Chap.4.2].

(o + ∂μ Γρμρ + Γμρλ Γλμρ) qνa = 0 .                                                 (4.16")

Evidently Evans' Wave Equation (4.16") cannot be true.




3. Evans' Chap.3: A Generally Covariant Field Equation ...

Evans "multiplies" Einstein's Field equation

(3.1)                                 R μν − ½ R gμν = T μν

by qνb ηab to obtain

(3.2)                                 Raμ − ½ R qaμ = Taμ

and suppresses the tetrad index a:

Rμ − ½ R qμ = Tμ                                                 (3.18)

That leads to Evans' "2nd Newton Law":

f μ = Tμ/∂τ                                                                 (3.22)

where τ is the proper time.

However, (neglecting other counterarguments e.g. concerning the "proper time") written with the suppressed index a we have

(3.3)                                 f μ = Taμ/∂τ                 (a = 0, 1, 2, 3),

that are four force-vectors instead of one in GR.

Which of them is the correct one? Evans gives no reply.
Evans' Equ.(3.22) contains a hidden type mismatch.


There is a further application of Equ.(3.2) or Evans' Equ.(3.18): We "wedge" it by qbν to obtain

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

and suppress the tetrad indices a,b

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

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

Evans remarks the term Rμ Ù qν being antisymmetric like the electromagnetic stress tensor Gμν . Hence he feels encouraged to try the following ansatz

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

where

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

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

(3.5)                                 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. (3.5) 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

(3.6)                                 qaμ Ù qbν ηab = qaμ qbν ηabqaν qbμ ηab = gμνgνμ = 0
and
(3.7)                                 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:
(3.8)                                 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.


References

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

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

[2]     G.W. Bruhn, Remarks on the "Evans Lemma" ;
         http://www.mathematik.tu-darmstadt.de/~bruhn/EvansLemma.html

[3]     W. A. Rodrigues Jr. and Q.A.G. de Souza, An ambigous statement called
         the "tetrad postulate"
, arXiv [math-ph/0411085]
         Int. J. Mod. Phys. D 12, 2095-2150 (2005)

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

[5]     G. W. Bruhn and W. A. Rodrigues Jr., Covariant Derivatives of Tensor Components,
         http://www.mathematik.tu-darmstadt.de/~bruhn/deblocking_dot.htm



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