#100(2) The Basic Field Equations

In the Maxwell Heaviside Theory the homogeneous equation is given by
                d Ù F = 0                                                 (1)
and the inhomogeneous equation by
                d Ù F^~ = μ_o J                                           (2)
in differential form notation.

Agreed so far.

These are first translated into tensor notation as follows. Eq.(1) is
                ∂_ρ F_μν + ∂_μ F_νρ + ∂_ν F_ρμ = 0 .                 (3)
and this is the same as
                ∂_μ F^{~ μν} = 0                                             (4)
where the tilde denotes Hodge dual
                F^{~ μν} = ½ Î^μνρσ F_ρσ .                             (5)

. . .

Q. E. D.

These considerations are incorrect for several reasons. The correct version of equ.(3) is

                ∂_[ρ F_μν] + C_[μρ^σ F_ν]σ = 0

where the additional term C_[μρ^σ F_ν]σ is caused by the anholonomicity of the coframe θ^σ (α=0,1,2,3) [1, p.146 eq.(B.4.31)]. The anholonomicity is required for the orthonormality [2, p.88, eq.(3.114)] of the frame

                g(e_α,e_β) = η_αβ

which cannot be attained for holonomic coframes θ^α = dx^α. In other words: Evans' eq. (3) is valid only in the special case of orthonormal Cartesian coordinates x^α (α=0,1,2,3).

This can be affirmed by reading Evans' papernote #100(4) , eq. (4), where he explicitly assures that he is dealing with the Minkowskian case

                g_μν = g^μν = diag[1, −1, −1, −1]

merely, i.e. with flat spacetime.

In curved spacetime, however, the Î-tensor is variable [2, p.52, eq.(2.43)]. Hence the Leibniz rule would yield additional terms in the eqs. (7-10).

Therefore eq.(4), ∂_μ F^{~ μν} = 0 , is wrong in general space-time.