Remarks on Evans' paper #100 - Section 3.

G.W. Bruhn, Darmstadt University of Technology


(Quotations from Evans' papers are displayed in

With the introduction to his paper #100 . Evans announces a review of major themes of his ECE theory. We continue reviewing paper #100 with its section 3 finding a further collection of older and recent flaws of that "theory".

1. The field ansatz

We read on p.12 of Evans' paper #100/3 [1.3]:

. . . The Bianchi identity (5) and its Hodge dual (9) become the homogeneous and inhomogeneous field equations of ECE respectively. These field equations apply to four fundamental fields of force: gravitational, electromagnetic, weak and strong and can be used to describe the interaction of the fundamental fields on the classical level. For example the electromagnetic by making the fundamental hypothesis:

                A = A(o) q                                                                                                 (19)

where the shorthand (index-less) notation has been used. Here A represents the electromagnetic potential form and cA(o) is a primordial quantity with the units of volts, a quantity which is proportional to the charge, −e on the electron. The hypothesis (19) implies that:

                F = A(o) T                                                                                                 (20)

where F is shorthand notation for the electromagnetic field form. The homogeneous ECE field equation of electrodynamics follows from the Bianchi identity (5):

                d Ù F + ω Ù F = A(o) R Ù q                                                                     (21)

and . . .

Evans bad habit of suppressing indices hides the problem attached to the above ansatz (19)/(20). With written indices instead of Eqs.(19)/20) we have

                Aa = A(o) qa         (a = 0, 1, 2, 3)                                                              (19')

we recognize the potential form Aa to be 4-vector-valued. The electromagnetic field is 4-vector-valued as well:

                Fa = A(o) Ta                                                                                              (20')

Evans gives no hint about the relation of the vector valued field form Fa to the scalar valued classical field form F which is the usual object of experimental research. However, from former publications by Evans one can deduce the relation

                F = F1 + F2 + F3                                                                                     (20'')

valid for arbitrary time-dependant real field forms F on an 3-D Euclidean space, the form F0 missing. See, e.g. .

However, as can easily be seen, the relation (20'') is not Lorentz invariant and therefore not appropriate for defining a measurement procedure. Evans never mentioned or used the field form F0. For further objections see [5] .

Evans' next equation

                d Ù F + ω Ù F = A(o) R Ù q                                                                     (21)

is a direct consequence of Evans' ansatz (20) and thus subject to the objections mentioned above. The following equation makes use of Evans' wrong conclusions about the Hodge dual of the 1st Bianchi identity in [1.2, between Eqs.(9) and (12)]:

                d Ù F~ + ω Ù F~ = A(o) R~ Ù q                                                                 (22)

as was pointed out in [6, Sect.1]. Evans inference

Therefore the ECE field equations are duality invariant, a basic symmetry which means that they transform into each other by means of the Hodge dual {1-12}.

is wrong therefore. His assertion would mean that only one of the equations (21) and (22) must be fulfilled, then the other one would be fulfilled automatically. This is nonsense well-known to all who ever worked with the pair of the Maxwell equations:

                d Ù F = 0                                                                                                 (23)

                d Ù F~ = J~o                                                                                         (24)

Since one of them is homogeneous (Eq.(23)) so - following Evans' assertion - should be the other one (Eq.(24), i.e. J~ = 0, which is evidently wrong. The reader should remember that at least in case of Minkowski spacetime we have ω = 0.

Therefore Evans' inference of Eq.(25),

                d Ù F = J/Îo = A(o) (R Ù q − ω Ù T)                                                        (25)

the equation

                d Ù F~ = J~o = A(o) (R~ Ù q − ω Ù T~)                                                 (26)

is invalid.

. . . The concise tensorial expression of the equations (25) and (26) is in general {1-12}:

                Dμ F~ aμν = A(o) R~ aμμν                                                                           (28)


                Dμ Faμν = A(o) Raμμν                                                                                 (29)

Both equations have (due to summation over the index μ) 2-tensors on both sides, while the tensor equivalents of the Eqs.(25)/(26) must equate 4-tensors. Thus, the eqs. (29) and (28) cannot be the tensor duals of the Eqs.(25)/(26) respectively. Evans' further conclusions are irrelevant because of being based on wrong assumptions described before.

2. The ECE wave equation

We read on p.19 of Evans' paper #100/3 [1.3]:

The ECE wave equation was developed {1-12} from the tetrad postulate {13}:

                Dμ qaν = 0                                                                                                       (44)

via the identity

                Dμ (Dμ qaν) = 0                                                                                               (45)

OK so far. But now caution! What follows now is a typical Evans-sophism, cf [7]:

This was reexpressed as the ECE Lemma

                o qaλ = R qaλ                                                                                                 (46)

This "Lemma" asserts the existence of a scalar R that satisfies 4×4 = 16 independent equations (46) (a,λ = 0,1,2,3). And Evans even announces the value of that scalar R:

                R = ¼ qλaμνμλqaν − ωaμbqbλ)                                                                 (47)

Now for checking Evans' Lemma we would have to insert the R given by Eq.(47) into Eq.(46). Dr. Evans does so applying his New Math in a blog note [1c]: He attempts to resolve Eq.(47) for the expression Γνμλqνa − ωaμbqλb :

We read and correct additional Evans' errors in red:

. . . i.e.

                o qαa = ∂μνμλqνa − ωaμbqλb)                                                       [1c, Eq.(8)]

Now define

                R = ¼ qaλμνμλqνa − ωaμbqλb)                                                 (47) / [1c, Eq.(9)]

and use

                qλaqaλ = 4                                                                                       [1c, Eq.(10)]

to find by using the eqs. [1c, Eq.(8-10)]

                o qλa =[1c, Eq.(8)]μνμλqνa − ωaμbqλb)
                                =[1c, Eq.(10)] ¼ [qλaqaλ] · ∂μνμλqνa − ωaμbqλb)
                                =(wrong) qλa · qaλμνμλqνa − ωaμbqλb)]
                                =[1c, Eq.(9)]
R qλa                                                             [1c, Eq.(11)]

The error =(wrong) in [1c] consists in an inadmissible use of summation indices. Evans' conclusion would not be possible if he would correctly avoid the indices a and λ in Eq. [1c,(10)], e.g. by replacing it with the equivalent equation (the reader should try it)

                qλbqaσ = 4 .


[1.1] M.W. Evans, A Review of Einstein-Cartan-Evans (ECE) Field Theory (Introduction of Paper #100), .

[1.2] M.W. Evans, Geometrical Principles (Section 2 of Paper #100:
      A Review of Einstein-Cartan-Evans (ECE) Field Theory
, .

[1.3] M.W. Evans, The Field (Section 3 of Paper #100:
      A Review of Einstein-Cartan-Evans (ECE) Field Theory
, .

[1.4] M.W. Evans, Aharonov Bohm and Phase Effects in ECE Theory (Section 4 of Paper #100:
      A Review of Einstein-Cartan-Evans (ECE) Field Theory
, .

[1.5] M.W. Evans, Tensor and Vector Laws of Classical Dynamics and Electrodynamics (Section 5 of Paper #100) , .

[1.6] M.W. Evans, Spin Connection Resonance (Section 6 of Paper #100) , .

[1a] M.W. Evans, Development of the Einstein Hilbert Field Equation . . ., .

[1b] M.W. Evans, Proof of the Hodge Dual Relation, .

[1c] M.W. Evans, Some Proofs of the Lemma, .

[1d] M.W. Evans, Geodesics and the Aharonov Bohm Effects in ECE Theory, .

[2] S.M. Carroll, Lecture Notes on General Relativity,, 1997.

[3] S.M. Carroll, Spacetime and Geometry,, 1997.

[4] F.W. Hehl and Y.N. Obukhov, Foundations of Classical Electrodynamics, Birkhäuser 2003

[5] G.W. Bruhn, Consequences of Evans' Torsion Hypothesis,
      ECEcontradictions.html .

[6] G.W. Bruhn, Remarks on Evans' paper #100 - Section 2,
      onMwesPaper100-2.html .

[7] M.R. Spiegel, Vector Analysis,
      in Schaum's Outline Series, McGraw-Hill.

[8] G.W. Bruhn, Evans' "3-index Î-tensor" ,
      Evans3indEtensor.html .

[9] G.W. Bruhn, Comments on Evans' Duality,
      EvansDuality.html .

[10] G.W. Bruhn, F.W. Hehl, A. Jadczyk , Comments on ``Spin Connection Resonance
      in Gravitational General Relativity''
, ACTA PHYSICA POLONICA B Vol. 39/1 (2008)
      pdf . html

[11] G.W. Bruhn, Remarks on Evans/Eckardt’sWeb-Note on Coulomb Resonance,,
      RemarkEvans61.html .


(08.01.2008) An Editorial Note by G. 't Hooft in Found. Phys.

(29.01.2008) Remarks on Evans' Web Note #100-Section 7: The Sagnac Effect

(25.01.2008) Remarks on Evans' Web Note #100-Section 6: SCR

(16.01.2008) Remarks on Evans' Web Note #100-Section 5: EM field

(08.01.2008) Remarks on Evans' Web Note #100-Section 4: The Aharonov Bohm effect

(05.01.2008) Remarks on Evans' Web Note #100-Section 3: Field and Wave equation

(01.01.2008) Remarks on Evans' Web Note #100-Section 2: Torsion and Bianchi identity

(27.12.2007) Remarks on Evans' Web Note #103

(19.12.2007) Myron now completely confused

(14.12.2007) Evans' Central Claim in his Paper #100

(10.12.2007) How Dr. Evans refutes the whole EH Theory