How Dr. Evans refutes the whole EH Theory

Gerhard W. Bruhn, Darmstadt University of Technology

In the Abstract of his web-paper [1] M.W.Evans proudly announces

It is shown that the use of the Christoffel connection in general relativity is inconsistent with the Bianchi identity of differential geometry. This finding means that the Einstein Hilbert (EH) theory is not a valid theory of physics . . .

With his eqs.[1,(28)-(29)] Evans arrives at a remarkable realization:

Christoffel connection:

                Γλμν = Γλνμ .                                 [1,(28)]

However, for the connection [1,(28)] the torsion must be zero:

                Tλμν = Γλμν − Γλνμ = 0.                 [1,(29)]

Therefore the Bianchi identity and Christoffel connection are incompatible in general. The Christoffel connection can be a solution of the Bianchi identity if and only if the metric defines a Ricci flat space-time where all elements of the Ricci tensor vanish.

There is no way in which the EH theory can escape its inherent weaknesses. Furthermore it is almost entirely based on the use of the Christoffel symbol, which as we have shown, violates the fundamental geometry [1,(1)]. It is concluded that all cosmologies based on EH theory are physically meaningless . . . The Christoffel symbol can be used if and only if

                Rκμμν = 0.                                 [1,(30)]

The reader is kindly requested have a look at the eqs. (3.80-95) of Evans' standard text book [2]. He'll find that at least S.M. Carroll (and a tiny minority of some other mislead guys) does not follow the above conclusion that vanishing torsion implies vanishing Ricci tensor.

The opinion of that tiny minority: In torsion free case due to Ta = 0 the 1st Bianchi identity D Ù Ta = Rab Ù qb yields Rab Ù qb = 0, i.e.

                Rλρμν + Rλνρμ + Rλμνρ = 0 ,                                 [1,(32)]

for short Rλ[ρμν] = 0, however, that does NOT imply the vanishing of the Ricci tensor.

S.M. Carroll [2, eqs.(3.100)ff.] considers the two-sphere as his favorite (counter-)example: In [2, eqs.(3.101)] the Christoffels are displayed to show the vanishing of torsion. Then Carroll calculates the Riemann tensor [2, eqs.(3.102-103)] finally followed by the Ricci tensor in [2,eqs.(3.104)]. For everybody who has eyes to see:

The Ricci tensor in [2, eqs.(3.104)] is NOT ZERO.


[1] M.W. Evans,The Incompatibility of the Christoffel Connection with the Bianchi Identity,

[2] S.M. Carroll, Lecture Notes on General Relativity,, 1997.

[3] W.A. Rodrigues, Differential Forms on Riemannian (Minkowskian) and Riemann-Cartan Structures
    and some Applications to Physics
, arXiv


(27.12.2007) Remarks on Evans' Web Note #103

(19.12.2007) Myron's New Questionable Developments of Cartan Geometry

(14.12.2007) Evans' Central Claim in his Paper #100

(20.11.2007) Remarks on Evans' papernotes #100