Extended on Jan 29 and on Feb 22, 2007

Some Confusion in Evans' "Cartan Geometry"

Gerhard W. Bruhn, Darmstadt University of Technology

1. Application of exterior covariant derivatives

Let's start with some well-known basics of Cartan geometry: The moving frame compatibility relation

(C)                 ∂μ qνa − Γμλν qλa + ωμab qνb = 0         |         dxμ Ù dxν

gives rise to introduce the 1-forms

qa := qνa dxν         and         ωab := ωμab dxμ

[2, (J.36-37)], [4, p.239 (5.14)]. The forms qa are vector-valued. However, note that the forms ωab are not tensorial since under LLT's they transform inhomogeneously [1, (3.134)], [2, (J.15)], [4, p.230 (1.11)].

We have

d Ù qa + ωab Ù qb = qλa Γμλν dxμ Ù dxν = 2 qλa Γλν] dxμ Ù dxν = qλa Tμλν dxμ Ù dxν
= Tμaν dxμ Ù dxν =: Ta ,

thus, as a direct consequence of (C) we obtain [1, (3.137)], [2, (J.28)]

(C1)                 Ta = d Ù qa + ωab Ù qb .         (1st Maurer-Cartan structure equation)

Application of dÙ to (C1) yields

                                d Ù Ta = 0 + d Ùac Ù qc)
                                = (d Ù ωac) Ù qc − ωab Ù (d Ù qb)
                                = (d Ù ωac) Ù qc − ωab Ù (Tb − ωbc Ù qc)
                                = (d Ù ωac + ωab Ù ωbc) Ù qc − ωab Ù Tb


                                d Ù Ta + ωab Ù Tb = (d Ù ωac + ωab Ù ωbc) Ù qc .

Introducing here [1, (3.138)], [2, (J.29)], [4, p.240 (6.10)]

(C2)         Rab := d Ù ωab + ωac Ù ωcb     (2nd Maurer-Cartan structure equation)

we obtain [1, (3.140)], [2, (J.31)]

(C3)                 d Ù Ta + ωab Ù Tb = Rab Ù qb .         (Ricci, 1st Bianchi identity)

Applying dÙ to (C2) yields

                                d Ù Rab = 0 + d Ùac Ù ωcb) = (d Ù ωac) Ù ωcb − ωac Ù (d Ù ωcb)
                                             = (Rac − ωad Ù ωdc) Ù ωcb − ωac Ù (Rcb − ωcd Ù ωdb) ,


(C4)                 d Ù Rab = − ωcb Ù Rac − ωac Ù Rcb .         (2nd Bianchi identity)

We introduce the exterior covariant derivative D Ù of p-forms Xa, Xb and Xab respectively ([1, (3.136)], [2, (J.31)]) by

                        D Ù Xa := d Ù Xa + ωac Ù Xc ,
                        D Ù Xab := d Ù Xab + ωac Ù Xcb + ωcb Ù Xac

to obtain for the 2-form Rab [3, (19.24a)]

(C4')             D Ù Rab = 0 .                 (2nd Bianchi identity)

(C1) can be rewritten as

(C1')             D Ù qa = Ta .                 (1st Maurer-Cartan structure equation)

Applying (D) to (C2) yields

                Rab := d Ù ωab + ωac Ù ωcb = d Ù ωab + (ωac Ù ωcb − ωcb Ù ωac) + ωcb Ù ωac


(C2')         D Ù ωab = Rab + ωac Ù ωcb .         (2nd Maurer-Cartan structure equation)

(D) applied to (C3) yields [3, (19.24a)]

(C3')             D Ù Ta = Rab Ù qb .                                 (Ricci, 1st Bianchi identity)

Remark. Evans writes the 2nd Bianchi identity in the shorthand form (C4') which means that he used Eq.(D) in that case.

            D Ù Rab = d Ù Rab + ωac Ù Rcb + ωcb Ù Rac := 0                 [5, (D.1)]

However, he also writes the 2nd Maurer-Cartan equation (C2) in shorthand notation [5, (7.2) + (7.4) + (9.60) + entries #28,#31 on p.187 + (16.4) + (17.16)* + (17.26) . . . ] e.g.

            Rab = D Ù ωab = d Ù ωab + ωac Ù ωcb                                 [5, (17.16)]

evidently now assuming another meaning of the "exterior covariant derivative".

(D')             D' Ù Xab = d Ù Xab + ωac Ù Xcb .

Evans' readers are warned about the possible confusions that could arise from the inconsistency (D)//(D') of tacitly using two different meanings of DÙ in [5].

The question whether D'Ù can be considered as covariant is anwered by Arkadiusz Jadczyk in [6].

2. Evans' Poincaré Lemma

The usual Poincaré Lemma

(PL)                 d Ù d = 0

is valid due to the commutativity of partial differential operators ∂μ, ∂ν . In [5, (9.68)] Evans extends this Lemma to exterior covariant derivatives as given by (D):

                D Ù D = 0                                                 [5, (9.68)]

So we'll check this Evans assertion by calculating (D Ù D) Ù Xa := D Ù (D Ù Xa) using the definition (D) of exterior covariant derivatives:

                D Ù (D Ù Xa) = d Ù (D Ù Xa) + ωac Ù (D Ù Xc)
                = d Ù (d Ù Xa + ωab Ù Xb) + ωac Ù (d Ù Xc + ωcb Ù Xb)
                = 0 + d Ùab Ù Xb) + ωac Ù (d Ù Xc) + ωac Ù ωcb Ù Xb
                = (d Ù ωab + ωac Ù ωcb) Ù Xb

hence, using (C2)

(DD)                 (D Ù D) Ù Xa = Rab Ù Xb .

Thus, Evans' Poincaré rule [5, (9.68)] is valid if and only if the curvature forms Rab are vanishing, i.e. only on flat spacetime.

3. ECE Theory and Cartan Geometry

Evans idea is to interprete geometric properties of spacetime as the reasons of electrodynamics in a certain analogy to Einstein who interpreted curvature of spacetime as the source of gravitation. So what remains from geometry is the torsion tensor T of spacetime to be brought in connection with the electromagnetic field tensor G.

On first view there are striking similarities: Both tensors are antisymmetric, and both can be derived from potentials.

(3.1)                 Geometry:         T = D Ù q         and         Electrodynamics:        G = D Ù A

Following Evans' (bad) habit of suppressing indices it is suggesting to assume proportionality between the fields G and T and between the potentials A and q.

(3.2)                 G = A(o) T         and         A = A(o) q

where A(o) is some constant. And this is just that what Evans does.

Now we shall restore Evans' suppressed indices (cf Sec 1) to obtain

(3.1')                 Geometry:         Ta = D Ù qa         and         Electrodynamics:        G = D Ù A
                (1st Maurer-Cartan structure relation)

(3.2')                 G = A(o) Ta         and         A = A(o) qa .

where a = (0),(1),(2),(3), i.e. a can attain four possible values due to geometric reasons, while due to physical reasons for the electrodynamic quantities (as every experimentalist will assure at least for the vacuum case) there exists only one field tensor G and one electromagnetic potential A. Since the transformation behavior of tensors is indicated by their indices it is very dangerous and misleading to suppress indices. So by restoring the hidden indices we have

The Eqs. (3.1'-2') contain a type mismatch: Both sides have different size of coefficient schemes and different behavior under local Lorentz transforms.

Possibly, Evans knows about this problem too. Therefore he assumes that the electromagnetic potentials and the corresponding fields consist of three orthogonal components

(3.3)                 Aa         and         Ga ,         (a = (1),(2),(3))

declaring that the actual fields (in case of free spacetime at least) are given by the sum of the components

(3.4)                 A = A(1) + A(2) + A(3) ,         G = G(1) + G(2) + G(3) ,

with missing zeroth components.

Hence Evans assumes the existence of a four-covector Ca such that

(3.5)                 A = Ca Aa ,         G = Ca Ga

is the actual potential and the actual field respectively. Since all tetrads can be transformed mutually into each other by local Lorentz transforms (LLTs) they are all equivalent: Hence the "composition" covector C = (Ca) must be the same for all these tetrads. The Eqs.(3.4) yield

(3.6)                 (Ca) = (0,1,1,1) .

On the other hand, the covector C = (Ca) must transform contravariantly, which is obviously violated by Equ.(3.6). Thus, we have:

Evans' assumptions (3.4-6) don't transform covariantly under LLTs and are invalid therefore.

Conclusion: No further theory can be built upon that dubious basis.

ECE Theory is obsolete.


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

[2] S.M. Carroll, Spacetime and Geometry, Addison Wesley 2004

[3] E. Zeidler in Teubner Taschenbuch der Mathematik Teil II, 8.Aufl. 2003

[4] Y. Choquet-Bruhat, Géométrie différentielle et systèmes extérieurs, Dunod Paris 1968


[6] A. Jadczyk, Remarks on Evans' "Covariant" Derivatives,