Remarks on Evans' 2nd Bianchi Identity

Gerhard W. Bruhn, Darmstadt University of Technology

The 2nd Bianchi identity is an essential step in proving energy conservation from Einstein's extended field equation

(1)                 Rμν − ½ R gμν = Σμν

where Rμν is the Ricci tensor, R is the scalar curvature and Σμν denotes the energy-momentum-tensor. The 2nd Bianchi identity runs as follows

(2)                 DλRρσμν + DρRσλμν + DσRλρμν = 0 ,

derived under the assumption of zero torsion. In [1, p.81, (3.88)] and [2, p.128, (3.140)] we find the remark:

For a general connection there would be additional terms involving the torsion tensor.

However, M.W. Evans knows more: In his GCUFT book [3, p.325,(D.8)]and in [3a, (D.8)] we find:

                DρRασμν + DμRασνρ + DνRασρμ = 0 .                                 (D.8)

. . . The second Bianchi identity is true for ANY gamma connection.

Who is right?

We check Evans' calculation. In [3, App. D] Evans starts from

                DÙRab = dÙRab + ωac ÙRcb + ωcb ÙRac = 0                     (D.1)

which can be found also in [1, (3.141)] and [2, (J.32)] and can hence be considered as confirmed. However, the following equation

                DρRabμν = ∂ρRabμν + ωaρc Rcbμν + ωcρb Racμν                 (D.3)
                                et cyclicum.

has evidently missing terms since due to the rules of covariant differentiation each index should produce an additional term besides the partial derivative, but there exist no terms corresponding to the indices μ and ν. The sum

(3)                 − Γραμ Rabαν − Γραν Rabμα

is missing which under cyclical summation most likely gives rise to terms containing the torsion tensor.

Thus, Evans' 2nd Bianchi identity (D.8) is invalid in case of non-vanishing torsion.

Evans in [3, p.302], [3a, (D.8)]:

In contrast, note carefully that the second Bianchi identity (17.4) (= (D.8)) is ALWAYS true for any type of connection, because it is fundamentally the cyclic sum of commutators of covariant derivatives [2]:

                [[Dλ,Dρ],Dσ] + [[Dρ,Dσ],Dλ] + [[Dσ,Dλ],Dρ] := 0.                 (17.9)/(D.9)

Equ. (17.9) is true but NOT so its application to Rabμν as performed by (D.3) and yielding the invalid eqns. (D.8) and (17.4).


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

[2] S.M. Carroll, Spacetime and Geometry, Addison Wesley 2004


[3a] M.W. Evans, The Spinning and Curving of Spacetime ..., Preprint