 ## New Concepts from the Evans Unified Field Theory. Part One

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

Quotations in black from the Evans paper  (also contained in ):

. . . The Einstein field equation

Rμν − ½ R gμν = kTμν                                                                 (11)

. . . can be deduced [1-10] as a special case of the Evans field equaton.

Rμν = Rμa qνb ηab                                                                 (13)

Tμν = Tμa qνb ηab                                                                 (14)

gμν = qμa qνb ηab                                                                 (15)

together with the reversed equations

Rμa = Rμν qbν ηab                                                                 (13')

Gμν = Gμa qνb ηab                                                                 (13")

Gμa = Gμν qbν ηab                                                                 (13''')

Tμa = Tμν qbν ηab                                                                 (14')

qμa = gμν qbν ηab                                                                 (15')

qaμ = gμν qνb ηab                                                                 (15'')

In order to derive Eqs. (4) and (5), start from the Evans field Eq. (1) and use the following relations:

Gμa = − ¼ R qμa ,                                                                 (16)

Tμa = ¼ T qμa .                                                                 (17)

Equations (16) and (17) are derived from the definitions [1-12] of R and T introduced originally by Einstein :

R = gμν Rμν ,         T = gμν Tμν .                                                 (18)

Using the Einstein convention 

gμνgμν = 4                                                                                 (19)

and the Cartan convention 

qμa qaμ = 1,                                                                                                 (20)

Unknown and wrong "convention"! Correct is

qμa qaμ = 4 ,                                                                                                 (20')

not due to Cartan but due to the fact that the matrices (qμa) and (qaμ) are mutual inverses and Equ. (20) gives the trace of the 4×4 unit matrix.

together with the definitions (13) and (14), we obtain:

R = gμν Rμν

= qaμ qbν ηab Rμa qνb ηab

= (ηabηab)(qbν qνb) (qaμ Rμa) = 4 qaμ Rμa .                                                 (21)

### This calculation is wrong due to total ignorance of the rules of tensor calculus.

The correct calculation is

R = gμν Rμν =(15'') qaμ ηab qbν Rμν =(13') qaμ Rμa                                         (21')

The next erroneous step follows immediately:

Multiply either side of Eq. (21) by qμa to obtain

Rμa = ¼ Rqμa ,                                                                                                 (22)

Gμa = Rμa − ½ R qμa = − ¼ R qμa ,                                                                 (23)

which is Eq. (16). Similarly, we obtain Eq. (17).

This instruction "multiply" is not feasible since the indices μ and a are not free. The result, the eqs. (22) and (23), are obviously wrong since asserting the proportionality of the matrices (Rμa), (Gμa), (Tμa) to (qμa), which is equivalent to the proportionality of the matrices (Rμν), (Gμν), (Tμν) to (gμν). So there is no logical way to the aimed equations (22) and (23) which are identical with Eq.(16).

Herewith Evans' calculation has completely broken down.

Remark
On the basis of the equations (13)-(15) the Einstein field equations (11) are trivially equivalent to the Evans Field equations:

Rμa − ½ R qμa = k Tμa | · ηabqνb     Û     Rμν − ½ R gμν = k Tμν | · ηabqbν.

. . .

The starting point for this class of wave equation are the Einsteinian definitions 

R = Rμν gμν ,                                                                                                 (46)

T = Tμν gμν .                                                                                                 (47)

Multiplication on both sides by gμν gives

R gμν = 4 Rμν, T gμν = 4 Tμν,                                                                 (48)

i.e.,

Rμν = ¼ R gμν, Tμν = ¼ T gμν.                                                                 (49)

### References

 M.W. Evans, New Concepts from the Evans Unified Field Theory. Part One, FoPL 18, p.139 ff.
http://www.aias.us/documents/uft/a12thpaper.pdf

 M.W. Evans, Chap.14 of GCUFT Vol.1 Aramis 2005