 ## Comment on Evans' ''DEFINITIVE PROOF 1: ANTISYMMETRY OF CONNECTION.''

March 06, 2009

### 1. Evans' ''proof''

Referring to the commutator [Dμ,Dν] is superfluous: The relevant definition of the torsion tensor is correctly given by Eq. (4):

Tλμν = Γλμν − Γλνμ                                                 (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 Γλμν 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

Tλμν = − Tλνμ                                                 (5)

so

Γλμν = − Γλνμ                                                 (6)

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

### 2. Further remarks

(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 Tλμν = 0 and Rλσμν = 0 if μ=ν, which is a well knon consequence of the antisymmetry of Tλμν and Rλσμν 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 falsly derive the vanishing of the complete curvature tensor Rλσμρ.
In an additional note Evans confirms our supposition: He believes that Rλσμμ is the ''complete curvature tensor''.

Evans: A symmetric connection means that the standard model of gravitational physics is obsolete because the curvature and torsion vanish and the gravitational field is always zero, reduction ad absurdum.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