by G.W. Bruhn, Darmstadt University of Technology

July 15, specified on July 17, 2008

After just having criticized Evans' New Math rules of tensor calculus I came across Evans' most recent paper #116(2). On first glance I saw the following calculation:

. . .

where

ω^{κ}_{μb} = q^{κ}_{a}
ω^{a}_{μb}
(20)

Finally use:

ω^{κ}_{μb} T^{~bμν} =
q^{λ}_{b}
q^{b}_{λ}
ω^{κ}_{μλ}
T^{~λμν} =
4 ω^{κ}_{μλ}
T^{~λμν}
(21)

The index λ appears four times on the right hand side while *at most twice* is
admissible. To write this flaw in ''slow motion'' and by means of brackets and avoiding
four times appearing of the index λ by means of primes:

ω^{κ}_{μb} T^{~bμν} =
(ω^{κ}_{μλ}
q^{λ}_{b})
(q^{b}_{λ'}
T^{~λ'μν}) = . . .

Now we can drop the brackets: Using
q^{λ}_{b}
q^{b}_{λ'} = δ^{λ}_{λ'} yields

ω^{κ}_{μb} T^{~bμν} =
ω^{κ}_{μλ}
δ^{λ}_{λ'}
T^{~λ'μν} = ω^{κ}_{μλ} T^{~λμν}
(21')

differing from Evans' result (21) by a factor 4. This implies that the factor 4 must be dropped in the eqs. (22) and (28). And I ask you:

However, someone who cannot distinguish between between 1 and 4 should better change his profession. Doing math is not his best ability.

Evans' main problem and *unproven* claim is the validity of the dualization of
the 1st Bianchi equation:

DÙT^{a} =
R^{a}_{b}Ùq^{b}
(1)
?Û?
DÙT^{~a} =
R^{~a}_{b}Ùq^{b}
(23)

What he correctly shows is *not the same* and hence useless:

D_{[μ} T^{a}_{νρ]} = R^{a}_{[μνρ]}
(4)
Û
D_{μ} T^{~aμν} =
R^{~a}_{μ}^{μν}
(10)

and (under unprovable assumption of the dualized Bianchi eq. (23))

D_{[μ} T^{~a}_{νρ]} = R^{~a}_{[μνρ]}
(24)
Û
D_{μ} T^{aμν} =
R^{a}_{μ}^{μν}
(27)

The gap (1)?Û?(23) cannot be closed. The transition from
R^{a}_{[μνρ]} to
R^{~a}_{[μνρ]}
would require a *common* Î factor which does not exist
due to the changing lower index positions.
I had pointed to that flaw already in a
former remark but without any effect on Dr. Evans. There are guys who do not learn from their
faults.