Remarks on Evans' Web Note #103

Gerhard W. Bruhn, Darmstadt University of Technology

We start with


The self-inconsistency is most clearly shown by considering the Hodge dual [1-11] of the one true Bianchi identity. Such a consideration leads to a simple tensor equation

                Dμ Tκμν = Rκμμν                                                                                 (1)

which links the covariant derivative of the torsion tensor Tκμν to the Ricci type tensor Rκμμν on the right hand side. Summation over repeated internal indices means that this is a two index tensor. Summation over repeated indices occurs also on the left hand side of Eq.(1) so the indices are balanced as required in tensor algebra.

On both sides is summed over the index μ yielding a two tensor. However, the tensor notation of both sides of the 1st Bianchi DÙT = RÙq must yield a four tensor

                D Tκμν] = Rκ[ρμν] .                                                                             (1')

the right hand side of which is displayed in Eq.(7) of this paper with lowered first index κ

                Rκρμν + Rκνρμ + Rκμνρ = 0                                                                 (7)

Thus, Eq.(1) is NOT the correct tensor representation of the 1st Bianchi identity.

The Ricci-type tensor on the right hand side of Eq.(1) was evaluated by computer algebra {1-11} using the Christoffel connection as defined in the standard model of general relativity:

                Γρλα = ½ gρλ (∂μgλα + ∂αgμλ − ∂λgαμ)                                                 (2)

and was found to be non-zero in general (papers 93 onwards of

Indeed, the incorrect expression on the right hand side of Eq.(1) (which is the Ricci tensor) is NOT zero in general while the correct epression given by the right hand side of Eq.(1') is vanishing for the torsion free case.

It vanishes only when the conventional Ricci tensor is assumed to be zero by construction − the so-called vacuum or Ricci flat solutions of EH.

OK. But once more: The Ricci tensor is incorrect at this place, to be replaced with the right hand side of Eq.(1'). And the vanishing of the right hand side of Eq.(1') does not imply the vanishing of the Ricci tensor Rκμμν .

S.M. Carroll (see copy) evaluates the Ricci tensor of the two-sphere of radius a for the Christoffel case. The Ricci tensor Eq.(3.104) does not vanish and - a forteriori - by contraction the scalar curvature, both non-vanishing.

Evans mentions the symmetry of the metric tensor in his Eq.

                gμν = gνμ                                                                                                 (3)

On p.6 after Eq.(7) he adds that

                Eq.(7) is true if and only the metric is symmetric.                                     (*)

I am wondering whether Dr. Evans is knowing that any locally Minkowskian metric is symmetric: From the tetrad relation

                gμν = qμbqνb ηab         where (ηab) := diag(−1, 1, 1, 1)                 [2,Eq.(3.119)]

due to the symmetry of the Minkowkian (ηab) we may deduce Eq.(3). However, any (locally Minkowskian) spacetime manifold with non-vanishing torsion is a counter example to Evans statement (*) where we have symmetry of metric while Eq. (7) fails to be fulfilled.

Evans' statement (*) is wrong.


[1] M.W. Evans, Development of the Einstein Hilbert Field Equation . . ., Web Note #103

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

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


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