Last update: 18.08.2007

see also
Evans' "3-index, totally antisymmetric unit tensor"

Comments on Evans' Duality

Gerhard W. Bruhn, Darmstadt University of Technology

(Comments in blue, quotes in black)


The terms "first and second Evans duality equations" firstly appear at [1, p.138] and were not explained before. While Hodge duality in 4D is defined by means of the well-known Levi-Civita Îabcd tensor with four indices [4, p.16] [4a, p.24] Evans defines his own different 4D "duality" by means of a 4D "totally anti-symmetric third rank unit tensor" Îabc. However, in spite of his busy efforts in [1, Chap.1 and App. A] he fails in defining such a tensor: As will be pointed out below, the "Evans Îabc tensor" given by the definition [1, entry #25 of Table 1 on p.186] does not exist in the 4D case. This shortcoming makes a lot of Evans' considerations in [1] useless as will be listed below. In addition some other errors and fatal conclusions will be noticed.
The monita listed below apply also to some Evans articles that appeared in FoPL during the past years.

p.138, also [7]

In the condensed notation of differential geometry the first and second Maurer-Cartan structure relations are, respectively:

        Ta = D Ù qa,                                                 (7.1)

        Rab = D Ù wab,                                             (7.2)

where Rab is the Riemann form and Ta the torsion form. The first and second Bianchi identities are the homogeneous field equations of unified field/matter theory and are, respectively:

        D Ù Ta  := 0,   (Typo? Read: ... = Rab  Ù qb)          (7.3)

        D Ù Rab := 0.                                              (7.4)

The first and second Evans duality equations state that there exist the Hodge duality relations

        Rab = Îabc Tc                                            (7.5)

        wab = Îabc qc                                            (7.6)

where Îabc is the appropriate Levi-Civita symbol in the well defined [4,5] orthonormal space of the tetrad.

It must be imposed that in contrast to Evans' remark this "Evans duality" is clearly distinct from the well-known Hodge duality which is defined by means of the 4 index Î tensor, while Evans has postulated the existence of an appropriate 3 index Î tensor for that purpose.

However, a third rank tensor Îabc cannot be defined in the 4D case. In spite of a somewhat misleading remark in [4, p.20]:

"Meanwhile, the three-dimensional Levi-Civita tensor Îabc is defined just as the four-dimensional one, although with one fewer index."

But the indices are running over 1,2,3 only (3D case). Besides, the above remark is missing in Carroll's book [4a].

Evans has extended the 3D existence of third rank Î tensors inadmissibly to four dimensions. The entry #25 in [1, Table 1 on p.186] says

        Îabc      = 1 if (a,b,c) even /      = −1 if (a,b,c) odd /      = 0 otherwise

That is NO DEFINITION in 4D, because (a,b,c) is no permutation of (0,1,2,3) and hence the attributes "odd" and "even" do not apply. We could find only two spots in his book [1] where the third rank Î tensor is used correctly, that is entry #89 in Table 1 on p.190, and Eq.(18.52) on p.338, both 3D cases.

The following example

        cBk = − ½ Îijk Gij ,         Ek = − G0k,                                 (2.103)
. . .
where all quantities are generally covariant.

shows Evans' problems of handling tensor algebra correctly: (We neglect the formally incorrect position of some indices.) The tensor Gij transforms covariantly in 4D. The indices of Gij are running over 0,1,2,3 while the indices of Îijk are restricted to 1,2,3. So Eq. (2.103) is correct only if we restrict all indices to the values 1,2,3. It is no correct equation in 4D. On the other hand, we have to apply 4D tensor algebra, and that doesn't fit together.

The tensor Îijk does not belong to 4D tensor algebra, and thus, all equations of 4D tensor algebra containing the 3D Î-tensor become dubious automatically.

A preliminary remark to the some of the following entries that refer to shortcomings in [1, Chap.9] and [9]:

The main topic of that chapter, the "Evans Lemma", is wrong due to one of Evans' numerous flaws of thinking as was shown in [11]. Therefore it might appear superfluous to mention other errors of his. However, the error type of "Evans duality" is remarkable since it appears at several spots of his book [1] as listed below. And perhaps he will respond in his unsurpassable manner as he once did replying to an email of mine that had passed his blockade by chance:

Well Gerhard you always say that any calculation is wrong. So if you say . . . is wrong you are simply behaving in the same old way. This is very boring. You have no credibility and so I request you not to send me any further absurd e mail, or any e mail. ... If you go on like this you will certainly get a haircut in the Tower. ... ... So why don't you do something useful and resign? British Civil List scientist.

p.173, also [9]

The antisymmetric metric is defined in general by the wedge product of tetrads:

        qcμν(A) = qaμ Ù qbν ,                                          (9.9)

The definition of c = c(a,b) is missing. Probably (cf the remarks on p.480 (D.13) below) Evans assumes the "definition" by

        qcμν(A) = Îcab qaμ Ù qbν                                      (9.9')

which is invalid due to the non-existence of Îcab.

Thus, no "antisymmetric metric" is defined.

The invalidity of Eq.(9.9) means that no "antisymmetric metric" with covariant transformation behavior can be defined by Evans' method.

p.177, also [9]

When one self-consistently includes the torsion tensor, electromagnetism becomes a theory of general relativity defined essentially by the wedge product of tetrads

        Tc = R qa Ù qb                                              (9.34)

The definition of c = c(a,b) is missing. Probably Evans assumes the "definition" by

        Tc = R Îcab qa Ù qb                                      (9.34')

which is invalid due to the non-existence of Îcab.

Thus, no relation of that kind between torsion T and the "antisymmetric metric" (9.9) can exist.

Due to several reasons:
(i) No "appropriate third rank Î-tensor" exists in 4D.
(ii) No "antisymmetric metric" (9.9) exists.
(iii) No quantity R (as asserted by the wrong "Evans Lemma") exists.

and by the Evans lemma and wave equation. The same theory [3-7] also describes gravitation, and the inter-relation between gravitation and electromagnetism. It can be seen from Eq. (9.34) that the torsion is a vector valued two form of differential geometry [2]. The fact that it is vector valued means that the internal index c must have more than one (???) component, and this means that electromagnetism cannot be a U(1) gauge field theory.

p.181, also [9]

        − R qcμν(A) = kT qaμ Ù qbν ,                                      (9.58)

The definition of c = c(a,b) is missing. Probably Evans assumes the "definition" by

        − R qcμν(A) = kT Îcab qaμ Ù qbν                                 (9.58')

which is invalid due to the non-existence of Îcab.
Besides, Eq.(9.58) is only the "index contracted Einstein equation" R = −kT multiplied by the invalid definition of the "antisymmetric metric" (9.9).

p.182, also [9]

The gauge invariant electromagnetic field is then

        Ga = D Ù Aa = g Ab Ù Ac,                                 (9.62)

The definition of a = a(b,c) is missing. Probably Evans assumes the "definition" by

        Ga = D Ù Aa = g Îabc Ab Ù Ac,                                 (9.62')

With that relation (9.62) Evans attempts to establish a connection between the electro-magnetic field tensor G and geometry, namely referring to his "antisymmetric metric" (the definition of which has failed as we saw above by the invalidity of Eq.(9.9)):

His axiom is the proportionality of the electromagnetic potential 1-form A to the tetrad 1-form qa.

A ~ qa

Since the 1-form qa is vector-valued, this assumption leads to the consequence of the 1-form A being vector-valued as well. Hence the em field tensor G must be vector-valued also, which means that instead of one electromagnetic field (E,B) the experimentalists should be confronted with four fields (Ea,Ba) (a=0,1,2,3) at once, this effect already observable under the conditions of free spacetime.

Here Evans' restriction of a = 1,2,3 must be remarked, since his method, his O(3) symmetry hypothesis, his "third rank" Î-tensor, does not admit the fourth index a=0.

Anyway, as we have seen above, Evans proposal fails due to formal reasons, and so there is no reason of getting upset. No experimentalist will need new measurement equipment due to Evans' findings:

Evans' Eq. (9.62) doesn't transform covariantly in 4D.

p.182, also [9]

. . . the cyclically symmetric equation of differential geometry

        D Ù qa = κ qb Ù qc , R = κ²,                                 (9.64)

The definition of a = a(b,c) is missing. Probably Evans assumes the "definition" by

        D Ù qa = κ Îabc qb Ù qc                                      (9.64')

which generalizes the B cyclic theorem of O(3) electrodynamics [8-12].

As was proven in two different ways [12] [13] the "B cyclic theorem of O(3) electrodynamics" does not behave covariantly under Lorentz transforms. Here we see that also the generalization to a general theory is impossible.

        D Ù Ta = 0         (Typo? Read: ... = Rab  Ù qb)         (9.66)
. . .

Poincaré lemma
        D Ù D = 0                                     (9.68)

Such a Poincaré lemma doesn't exist. Eq.(9.68) is true only in flat spacetime [11].

p.183, also [9]

In the development of the Evans field theory, novel and important duality relations of differential geometry have been discovered (9.1) between the torsion (T) and Riemann forms (R)

As remarked above there are no "Evans duality" relations since being introduced by a non-existing third rank Î tensor in 4D. In addition, there is a serious confusion of notation: In Sect. 9.1 (p.172) the term T denotes the "index contracted energy momentum tensor" while here (at p.183) T means the torsion. Later on (Chap.18) Evans denotes the torsion by τ, but without correcting the just mentioned error.

and (9.2) between the spin connection (ω) and tetrad (q). These are fundamental to differential geometry, to topology, and to generally covariant physics. They are proven from the fact that the tangent space is an orthonormal Euclidean space, so in this space there exists the relation [2-7] between any axial vector Va and its dual antisymmetric tensor Vκ:

        Va = Îabc Vbc,                                            (9.73)

Typo: Read it as Va = Îabc Vbc to be at least formally correct. However, Vbc cannot denote the Hodge dual of Va which must have 3 indices in 4D. So Eq.(9.73) is dubious at all.

where Îabc is the Levi Civita symbol or three-dimensional totally antisymmetric unit tensor (see Table 1).

The entry #25 of Table 1 on p.186 does not apply for 4D.

Raising one index gives

        Îabc = ηcd Îdbc ,                                            (9.74)
     (Typo: Read it Îabc = ηad Îdbc; but Îdbc undefined)

where ηad = diag(−1, 1, 1, 1) is the metric of the orthonormal space. The first Evans duality equation of differential geometry is

        Rab = κ Îabc Tc,                                            (9.75)
(???: Îabc undefined. Factor κ? Compare with Eq.(7.5) quoted above)

showing that the spin connection (another confusion! meant is the torsion T; see the introductory remarks on the top of this page) is dual to the Riemann form, and the second Evans duality equation of differential geometry is

        ωab = κ Îabc qc                                            (9.76)
(???: Îabc undefined. Factor κ? Compare with Eq.(7.6) quoted above)

showing that the spin connection is dual to the tetrad. The duality equations define the symmetry of the Riemann form because they imply

        Rab = R Îabcd qc Ù qd,                                   (9.77)

Of course, dubious as well: Evans combines here Eq. (9.34') (with lowered index a) with Eq.(9.75) to obtain

        Rab = κ R Îabc Îcde qd Ù qe,

and assuming the "rule" Îabc Îcde = Îabde we get Eq.(9.77) with exception of the superfluous factor κ. So κ must be removed: No problem for Dr Evans: Use Eq.(7.5) instead of Eq.(9.75), and now everything is fine: Complete confusion!

p.330 f.

. . . from which the unified field theory is developed are as follows:

        D Ù Va := d Ù Va + wab Ù Vb                          (18.2)

        D qa = 0                                                          (18.3)

        tc = D Ù qc                                                      (18.4)

        Rab = D Ù qab.                                                 (18.5)

These equations define the fundamental properties of any non-Euclidean spacetime with both curvature and torsion in terms of differential forms. Eqs. (18.4) and (18.5) are the first and second Maurer Cartan structure relations [2], defining respectively the torsion (tc) and Riemann or curvature (Rab) forms in terms of the tetrad and spin connection respectively. These four equations are inter-connected in free space by the recently inferred (WHERE?) and fundamental first and second Evans duality equations of differential geometry [3]–[17]:

        wab = −κ Îabc qc                                     (18.6)         (???: Îabc undefined)

        Rab = −κ Îabc τc                                      (18.7)         (???: Îabc undefined)

where κ is the wave number. The novel Evans duality equations were inferred in free space from the fact that the torsion and Riemann forms are both antisymmetric in their base manifold indices μ and ν, so that one must be the dual of the other in the orthonormal space of the tetrad [2]
That's no sufficient argument!

        Rabμν = −κ Îabc τcμν                                (18.8)         (???: Îabc undefined)


        Îabc = ηda Îdbc.                                       (18.9)         (a typo: Meant is Îabc = ... but this tensor is undefined in 4D.)

Here Îabc is the Levi Civita symbol (totally anti-symmetric third rank unit tensor) and where ηda is the metric in this orthonormal (orthogonal and normalized [2]) space. The torsion form, a vector valued two form with index c, is therefore dual to the curvature or Riemann form, a tensor valued two form anti-symmetric in its a, b indices [2].

Further appearances of undefined rank 3 Î tensors in 4D


        Ab = − g Ab Ù (Ac + g Îcde + g² Îcfgh AfAgAh + . . . )                                (17.53)       (also formally incorrect)

p.334         (18.26)

p.335         (18.27-28)

p.346         (19.9)

p.347         (19.10)

p.401         (23.20)

p.438         (26.24)

p.449 f.

A particular solution of Eq.(27.27) is:

        ωab = − κ Îabc Ù qc                               (27.28)

where Îabc is the Levi-Civita tensor in the flat tangent bundle spacetime.Being a flat spacetime, Latin indices can be raised and lowered in contravariant covariant notation and so we may rewrite Eq.(27.28) as:

        ωab = κ Îabc Ù qc                                   (27.29)


1) Duality: τc = Îacb Rcb(A)


gc(A) is the antisymmetric part of qab(A):

        gc(A) = ½ Îabc qab(A)                                   (D.13)

Typo? Formally incorrect. Read it as

        gc(A) = ½ Îabc qab(A)                                   (D.13')

However, no third rank LC-symbols exist in 4D, cf the above remarks on entry 25 of [1, Table 1 at p.186].

p.488         (F.10-11)

p.490         (F.32)

p.497         (G.23-24)

p.498         (G.26-29)

p.501         (H.7)

p.511         (J.6-7)

For O(3) electrodynamics we choose:
              ωab = − ½ κ Îabc qc                             (J.13)
in the structure relation
              D Ù qa = d Ù qa + ωab Ù qb               (J.15)

Proof For a=1: . . .
The indices b=0 and c=0 occur at summation: What is e.g. Îab0? And what about a=0???

The free space condition means that:
              ωab = − κ Îabc qc                             (K.16)
              Rab = − κ Îabc Tc                             (K.17)


1. M.W. Evans, GCUFT, Arima 2005

4. S. M. Carroll, Lecture Notes on General Relativity (University of California,
     Santa Barbara, graduate course on arXiv:gr-qe / 9712019 v1 3 Dec, 1997).

4a. S. M. Carroll, Spacetime and Geometry, Addison Wesley 2004.

5. R. M. Wald, General Relativity (Chicago University Press, 1984).

7. M.W. Evans, Unification Of Gravitational And Strong Nuklear Fields, FoPL 17 p.267 ff.

9. M.W. Evans, The Evans Lemma Of Differential Geometry, FoPL 17 p.433 ff.

10. G.W. Bruhn, A Fatal Error in M.W. Evans' EVANS Lemma

11. G.W. Bruhn, Some further confusion in Evans' "Cartan geometry"

12. G.W. Bruhn, No Lorentz property of M W Evans' O(3)-symmetry law, Phys. Scr. 74 (2006) 537–538

13. G.W. Bruhn, No Lorentz Invariance of M.W. Evans' O(3)-Symmetry Law,