Remarks on Evans' papernotes #100

G.W. Bruhn, Darmstadt University of Technology

#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~ = μo J                                           (2)
in differential form notation.

Agreed so far.

These are first translated into tensor notation as follows. Eq.(1) is
                ∂ρ Fμν + ∂μ Fνρ + ∂ν Fρμ = 0 .                 (3)
and this is the same as
                μ F~ μν = 0                                             (4)
where the tilde denotes Hodge dual
                F~ μν = ½ Îμνρσ Fρσ .                             (5)

. . .

Q. E. D.

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

                ∂ Fμν] + C[μρσ Fν]σ = 0

where the additional term C[μρσ Fν]σ is caused by the anholonomicity of the coframe θσ (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(eα,eβ) = ηαβ

which cannot be attained for holonomic coframes θα = dxα. In other words: Evans' eq. (3) is valid only in the special case of orthonormal Cartesian coordinates xα (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

                gμν = gμν = 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.

In addition, the factors ½ in the eqs.(7-9) are wrong which is, of course, of minor importance.

Analogous objections hold for the eqs. (11-12). The validity of the subsequent eqs. (13-16) was therefore shown only for the Minkowskian case where the equations are well-known.


[1] F.W. Hehl and Y.N. Obukhov, Foundations of Classical Electrodynamics, Birkhäuser 2003

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


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