 ## ECE Theory and Cartan Geometry

### Gerhard W. Bruhn, Darmstadt University of Technology

This report is Sec.3 of a preceeding article on 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 τ of spacetime to be brought in connection with the electromagnetic field tensor F. (Sometimes Evans writes G instead of F and T instead of τ.)

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

(3.1)                 Geometry:         τ = D Ù q         and         Electrodynamics:        F = D Ù A . [8, (28)]

Following Evans' (bad) habit of suppressing indices it is suggesting to assume proportionality between the fields F and τ and between the potentials A and q [5,(9.55),(9.61-62),(18.14-15)].

(3.2)                 F = A(o) τ         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 of the former article on Cartan-Geometry) to obtain

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

(3.2')                 F = A(o) τa         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 F 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         Fa ,         (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) ,         F = F(1) + F(2) + F(3) ,

with missing zeroth components [7, Chap.1,(1.1.5)] , [5, App.C p.475 f.] .

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

(3.5)                 A = Ca Aa ,         F = Ca Fa

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:

## Conclusion: No further theory can be built upon that dubious basis. ECE Theory is obsolete.

### References

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

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

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

 Y. Choquet-Bruhat, Gťomťtrie diffťrentielle et systŤmes extťrieurs, Dunod Paris 1968

 M.W. Evans, GENERALLY COVARIANT UNIFIED FIELD THEORY, Arima 2006

 A. Jadczyk, Remarks on Evans' "Covariant" Derivatives,