On the Hodge dual of the first Bianchi identity

Gerhard W. Bruhn, Darmstadt University of Technology

Dec 3, 2008

One of the basic equations of differential geometry is the 1st Bianchi identity

(1)         D Ù T^a = R^a_b Ù q^b .

In their webnote [1] the authors attempt to dualize that identity by claiming the Hodge dual of eq.(1) to be

(1^~)         D Ù T^a~ = R^a_b^~ Ù q^b

as an up to now unknown further basic equation of differential geometry.

To prove that new equation (1^~) is transformed to the tensorial equation

(2)         D_μ T^κμν = R^κ_μ^μν

which is considered to be equivalent to the dualized 1st Bianchi identity (1^~). Therefore the tensorial equation (2) should be valid for arbitrary sample manifolds of Riemannian differential geometry. We'll check this assertion (2) by an elementary example below. The reader will find a much deeper treatment of the topic by W.A. Rodrigues Jr. in [4].

1. The unit-2-sphere S² in R³

Using some basic informations from S.M. Carroll [3] we have the metric

(1.1)         ds² = dθ² + sin²θ dφ² = dx^{1 2} + sin² x¹ dx^{2 2}

with the metric tensors

(1.2)                 (g_μν) = diag( 1, sin² x¹) ,         (g^μν) = diag( 1, ¹/_{sin² x¹}) .

using the numbering of indices

(1.3)         1 ~ θ, 2 ~ φ

There are only a few non-vanishing Christoffel coefficients

(1.4)         Γ¹₂₂ = − sin x¹ cos x¹ ,     Γ²₁₂ = Γ²₂₁ = cot x¹ ,

while all other Christoffels Γ^κ_μν vanish.

The torsion T^κ is given by

(1.5)         T^κ_μν = Γ^κ_μν − Γ^κ_νμ = 0 ,

i.e. vanishing due to the symmetry of the Christoffels in their lower indices μ,ν.

2. The Riemann tensor of S²

The Riemann tensor is given by

(2.1)         R^κ_λμν = ∂_μΓ^κ_λν − ∂_νΓ^κ_λμ + Γ^κ_μρΓ^ρ_νλ − Γ^κ_νρΓ^ρ_μλ

being antisymmetric in μ,ν as is well-known. Therefore we have especially

(2.3)                 R^κ_λμν = 0     if     μ = ν.

3. The check

Due to the vanishing of torsion (1.5) the equation (2) to be checked reduces to

(3.1)         0 = R^κ_μ^μν = R^κ_μαβ g^μα g^νβ .

Therefore, due to the diagonal form (1.2) of (g^μρ), we have to check:

(3.2)         (R^κ_11β g¹¹ + R^κ_22β g²²) g^νβ = (R^κ₁₁₁ g¹¹ + R^κ₂₂₁ g²²) g^ν1 + (R^κ₁₁₂ g¹¹ + R^κ₂₂₂ g²²) g^ν2 .

This is:

for ν=1:     R^κ_μ^μ1 = (R^κ₁₁₁ g¹¹ + R^κ₂₂₁ g²²) g¹¹ + 0 = (R^κ₁₁₁ g¹¹ + R^κ₂₂₁ g²²) g¹¹ = R^κ₂₂₁ g²² g¹¹ ,

for ν=2:     R^κ_μ^μ2 = 0 + (R^κ₁₁₂ g¹¹ + R^κ₂₂₂ g²²) g²² = (R^κ₁₁₂ g¹¹ + R^κ₂₂₂ g²²) g²² = R^κ₁₁₂ g¹¹ g²² ,

i.e. the test reduces to:

(3.3)         R^κ₂₂₁ = 0 ?         and         R^κ₁₁₂ = 0 ?

We consider the special case κ = 1 to obtain:

(3.4)         R¹₂₂₁ = 0 ?         and         R¹₁₁₂ = 0 ?

The check 'R¹₂₂₁ = 0 ?' means in detail

(3.5)         R¹₂₂₁ = ∂₂Γ¹₂₁ − ∂₁Γ¹₂₂ + (Γ¹₂₁Γ¹₁₂ + Γ¹₂₂Γ²₁₂) − (Γ¹₁₁Γ¹₂₂ + Γ¹₁₂Γ²₂₂) = 0 ?
                                =o                             =o                               =o                 =o

thus

(3.6)         R¹₂₂₁ = − ∂₁Γ¹₂₂ + Γ¹₂₂Γ²₁₂ = − sin² x¹ ≠ 0 .

Therefore we have obtained a negative check result: The test equation (2) is not fulfilled for the unit-2-sphere S² which means:

Eq.(2) is invalid in general.

Remark: Another counter example to eq.(2) is given by the Schwarzschild metric in [2]: Sect.1.1.4 gives symmetric Christoffel connection, hence the torsion is zero. However, due to Sect.1.1.12 we have R^o_μ^μo ≠ 0, again contradicting eq.(2).

References

[1] M.W. Evans, H. Eckardt, Violation of the Dual Bianchi Identity by Solutions of the Einstein Field Equation
      Violation of the Dual Bianchi Identity

[2] M.W. Evans, H. Eckardt, Spherically symmetric metric with perturbation a/r
      http://www.atomicprecision.com/blog/wp-filez/a-r.pdf

[3] S.M. Carroll, Lecture Notes on General Relativity, p.60 f.,
      http://www2.mathematik.tu-darmstadt.de/~bruhn/Carroll84-85.bmp

[4] W.A. Rodrigues Jr., Differential Forms on Riemannian (Lorentzian) and
      Riemann-Cartan Structures and Some Applications to Physics
      Ann. Fond. L. de Broglie 32 424-478 (2008)
      http://arxiv.org/pdf/0712.3067

[5] G.W. Bruhn, Evans' Duality Experiments
      http://www2.mathematik.tu-darmstadt.de/~bruhn/Duality.html

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