## Consequences of Evans' Torsion Hypothesis

### Gerhard W. Bruhn

Motto by Lar Felker:

Evans shows electromagnetism is spacetime torsion; Einstein showed gravitation is spacetime curvature.
Cartan differential geometry is sufficient to mathematically describe GR and EM together.

### 1. The Hypothesis

Evans assumes an analogy of electrodynamics to geometry, as described e.g. in his GCUFT book [1,App.F].

The geometry is converted to the field theory using:

Aa = A(0) qa                                                                                                 (F.2)

Fa = A(0) Ta.                                                                                                 (F.3)

Here A(0) is the fundamental potential magnitude, with units of volt s/m, Aa is the potential form and Fa the field form.

### 2. The number of field forms

While in classical Maxwell theory only one scalar valued field form appears we have a 4-vector valued field form Fa (a=0,1,2,3) in ECE theory.

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                                                                                                                 (2.1)

valid for arbitrary time-dependant real field forms F on an 3-D Euclidean space, the form F0 missing. See, e.g.
http://www.atomicprecision.com/blog/2007/08/26/first-attempt-at-b-3-field-animation-ideas-3/.

However, as can easily be seen, the relation (2.1) is not Lorentz invariant and therefore not appropriate for defining a measurement procedure.

### 3. The Maxwell Equations in Free Space

The 1st Bianchi equation D Ù Ta = Rab Ù qb multiplied by A(0) yields due to Evans' torsion hypothesis (F.2-3)

D Ù Fa = Rab Ù Ab         or         d Ù Fa = Rab Ù Ab − ωab Ù Fb .                                                 (3.1)

To make that agree with the usual form of the homogeneous Maxwell equation d Ù Fa = 0 Evans introduces the concept of free space:

From eqn. (F.1) to (F.3) we obtain the homogeneous Evans field equations (HE):

d Ù Fa = Rab Ù Ab − ωab Ù Fb = − A(0) (− qb Ù Rab + ωab Ù Tb) .                                                 (F.4)

With the geometrical constraint <of free space>:

Rab Ù qb = ωab Ù Tb                                                                                                                 (F.5)

eqn. (F.4) reduces to the homogeneous Maxwell-Heaviside field equations:

d Ù Fa = 0                                                                                                                                 (F.6)

for each index a.

The constraint of free space is nothing but

d Ù Ta = Rab Ù qb − ωab Ù Tb = 0 .                                                                                                 (3.2)

However, from (3.2) follows:

### The constraint of free space is not Lorentz invariant, hence "free space" no physical property.

Remark: This holds for spacetime with torsion T. In torsion-free spacetime, i.e. for Ta=0, the free space condition is fulfilled identically.

Evans now attempts to explain the index a (=0,1,2,3):

The superscript a is the tangent bundle index of differential geometry, and physically indicates states of polarization.

"States of polarization" is a reference to Evans' cyclical basis ea (a=1,2,3) he had introduced in case of a circularly polarized plane wave. But he has never explained how the vectors ea (a=1,2,3) should be defined in case of more general electromagnetic waves. And a definition of a vector eo is totally missing.

At last Evans attempts to relate the concept of free space to a certain duality concept (of his, not Hodge! See [2,3]):

Using the structure equations:

Ta = D Ù qa = d Ù qa + ωab Ù qb ,         Rab = D Ù ωab = d Ù ωab + ωac Ù ωcb                 (F.8)

of differential geometry, eqn (F.5) becomes:

(D Ù ωab) Ù qb = ωab Ù (D Ù qb)                                                                                         (F.9)

one possible solution of eqn. (F.9) is:

ωab = − κ Îabc qc                                                                                                                 (F.10)

Rab =    κ Îabc Tc                                                                                                                 (F.11)

where κ is the wavenumber (which κ in case of a general wave?).

Eqns (F.10) and (F.11) mean that in free space, ωab is the antisymmetric tensor dual to the vector qc, and Rab is the antisymmetric tensor dual to Tc. These equations define free space electromagnetism free of gravitation influence (mass). In this case the spin connection and tetrad are duals and the curvature and torsion forms are duals.

The free space conditions (F.5) and (F.9) give rise to the problem of fulfilling them anyhow. Therefore Evans introduces a "duality" between the forms ω and q by eq.(F.10) and between R and T by eq.(F.11). However:

Therefore:

### The "duality" conditions (F.10-11) are null and void.

Nevertheless the non-existing 3-index Îabc tensors are used in [1,App.G] to calculate representations of the field forms Fa (a=1,2,3) from the potential forms Ac (c=1,2,3), see [1,G.29). It must be noticed that neither Fo nor Ao appear in that calculations. All index summations are running over the indices 1,2,3, the index 0 is missing:

F1 = d Ù A1 + g/2 (Î123 A3 Ù A2 + Î132 A2 Ù A3)
F1 = d Ù A1 + g A2 Ù A3                                                                                                                 (G.29)
F2 = d Ù A2 + g A3 Ù A1
F3 = d Ù A3 + g A1 Ù A2

Eqns. (G.29) are the fundamental definition of the field tensor of electromagnetism in objective physics.

In these equations:

g = κ/A(o)                                                                                                                                 (G.30)

and should not be confused with the determinant of the metric.

### 4. A Conclusion for Torsion-free Spacetime

The condition T=0 is just fulfilled for the class of all spacetimes considered in GRT. Due to Evans torsion hypothesis we have the conclusion:

### 5. A Last (But Not Least) Contradiction

Evans in http://www.atomicprecision.com/blog/2007/08/20/calculation-of-christoffel-symbols/

As we know, motion in galaxies is torsion dominated, but in the solar system, torsional effects are very weak.

However, according to the torsion hypothesis (F.3) one should expect high torsion at places with high em radiation, so e.g. in our solar system near the sun ...

### References

[1] M.W. Evans, Generally Covariant Unified Field Theory, the geometrization of physics, Arima 2006

[2] G.W. Bruhn, Evans' "3-index, totally antisymmetric unit tensor",
http://www.mathematik.tu-darmstadt.de/~bruhn/Evans3indEtensor.html

[3] G.W. Bruhn, Comments on Evans' Duality,
http://www.mathematik.tu-darmstadt.de/~bruhn/EvansDuality.html