''DEFINITIVE PROOF 1:

ANTISYMMETRY OF CONNECTION.''

March 06, 2009

Referring to the commutator [D_{μ},D_{ν}] is superfluous:
The relevant definition of the torsion tensor is correctly given by Eq. (4):

T^{λ}_{μν} =
Γ^{λ}_{μν} −
Γ^{λ}_{νμ}
(4)

See Carroll's Lecture Notes on General Relativity p.59, Eq.(3.16). This definition
guaranties that the torsion tensor is antisymmetric in the lower indices μ,ν,
**irrespective whatever the connection Γ ^{λ}_{μν} is.**
Two cases are of interest:

(a) The connection is symmetric: This is the case if and only if the torsion vanishes.

(b) The connection is asymmetric: This is the case if and only if the torsion is not zero.

Therefore the concluding ''so'' in

T^{λ}_{μν} = − T^{λ}_{νμ}
(5)
**
**

**so**

Γ^{λ}_{μν} = − Γ^{λ}_{νμ}
(6)

is the **point of flaw**: Where is the reasoning for that ''so''???

(1): Eq. (3) contains an index error at the first term on the right hand side. The lower index ρ
should be ν. A slip only? **But a slip with consequences**:

(2): In Eq. (2) the case μ=ν is specified: This yields the conclusion
T^{λ}_{μν} = 0 and R^{λ}_{σμν} = 0
if μ=ν, which is a well knon consequence of the antisymmetry of
T^{λ}_{μν} and R^{λ}_{σμν}
in μ,ν (see e.g. S.M. Carroll L.N. p.59, Eq.(3.16), p.75, Eq.(3.64)). After Eq. (3.16)
Carroll writes: ''*It is clear that the torsion is antisymmetric in its lower indices,
and a connection which is symmetric in its lower indices is known as ''torsion-free''.*''
However, due to Evans' ''slip'' in Eq.(3) one could

In an additional note Evans confirms our supposition: He believes that R

Evans:
*A symmetric connection means that the standard model of gravitational physics is
obsolete because the curvature and torsion vanish and the gravitational field is always
zero, reduction ad absurdum.*

(3): Evans' '' DEFINITIVE PROOF 2: THE FUNDAMENTAL ORIGIN OF TORSION AND CURVATURE'' starts with the following statements:

''*
The torsion and curvature tensors in general relativity are defined by the action of
the commutator of covariant derivatives on any tensor. This proof considers the action of
the commutator on the vector in any dimension and in any spacetime. The proof is true
irrespective of any assumption, even fundamentals such as metric compatibility or tetrad
postulate. The spacetime torsion is always present in general relativity and cannot be
ignored. This means that the connection is always antisymmetric , not symmetric as in the
standard model.
*

* Proof:*
''

It remains unclear what shall be proven in the following where Evans considers the appearance
of torsion and curvature tensors in the commutator applied to a (1,0)-tensor.
As shown above, however, it cannot be proven that
''*the connection is always antisymmetric*''.