Reviews of two Papers on B. Lehnert's "Extended Electromagnetic Theory"

Gerhard W. Bruhn, Darmstadt University of Technology

Corrected errors are displayed in red, quotations from the articles in black with equation labels at the right margin. Our own equation labels stand on the left margin.

1.   The Paper on "Extended Electromagnetic Theory ... " [1] by B. Lehnert and S. Roy


The reason for considering this article is the consequence of an unclear formulation in Sect.2.1 on another article where the same authors were co-authors together with a greater group of other authors. The unclear formulation is the role of the Lorenz condition for the derivation of Equ.(7). We trace the calculation until that equation and mark some minor typos in red. The consequences are discussed in the next review where the wrong interpretation of this equation (7) gives rise to miraculous conclusions on vacuum currents and vacuum energy. It's a pity that vacuum currents and vacuum energy have their origin in just the wrong interpretation of Equ.(7) as "Lorenz-free", a mere mix-up error.

The consideration starts with the well-known Maxwell equations

curl B1/Et = μo j                                                                 (1)

curl E + Bt = 0                                                                 (2)

Then div (1) with 1/ = εoμo yields div j + ρt = 0 . Let

B = curl A .                                                                 (5)

Then curl (5) yields

curl curl A = curl B = μoj + 1/ Et

or due to

E = − ÑΦ − At                                                                 (6)


curl curl A = μoj1/ (ÑΦ + At)t

or with curl curl A = grad div A − ΔA

[R;(7a)]                                 1/ Att− ΔA + Ñ (div A + 1/ Φt) = μo j .

An analogous equation can be derived for Φ: div (6) yields with

div E = ρ/εo                                                                 (4)

the relation

ρ/εo + div At + ΔΦ = 0


[R;(7b)]                                 1/ Φtt − ΔΦ − (div A + 1/ Φt)t . = ρ/εo .

Eqns. [R;(7a,b)] can be combined to

(1/ ∂²/∂t² − Δ) (A,Φ) = ( μoj , ρ/εo)                                                                 (7)

by before using the Lorenz*) condition

div A + 1/ Φt = 0 .                                                                 (8)

However, in the subsequent article [2] the authors believe in (7) being derived without assuming the Lorenz*) condition.

This has consequences for the article [2] reviewed next where both authors appear as co-authors. As far as I have checked other articles, e.g. [3] and [4] were not affected by the error in [2].

*) Remark on the Lorenz-Convention falsely attributed to H.A. Lorentz

The Lorenz condition stems from the Danish mathematician and physicist Ludvig Valentin Lorenz (18. Januar 1829 - 9. Juni 1891):

Lorenz, L. "On the Identity of the Vibrations of Light with Electrical Currents." Philos. Mag. 34, 287-301, 1867.

Further References

Wikipedia (in Danish)
Wikipedia (in German)
van Bladel, J. "Lorenz or Lorentz?" IEEE Antennas Prop. Mag. 33, 69, 1991.
Whittaker, E. A History of the Theories of Aether and Electricity, Vols. 1-2. New York: Dover, p. 268, 1989.

2.   The Paper on "Classical Electrodynamics without the Lorentz Condition: ... " [2] by M.W. Evans, T. Bearden e.a.


We trace the derivation of the differential equations for the vector potential from the homogeneous Maxwell equations for the vacuum and back. However, due to an unclear formulation in [1] for the way back the authors erroneously use the version [1; (7)], where the Loren(t)z condition was applied, instead of the "Lorenz-free" version [R;(7a)]. Therefore just this term now appears additionally as "vacuum current density". A miraculous conclusion,which should have made the authors suspicious, that just the Lorenz term should yield a vacuum current. Surely, it's a pity that vacuum currents and vacuum energy have their origin merely in that wrong interpretation of Equ.(7) as "Lorenz-free", and all further speculations for a vacuum energy density are in vain.

Abstract of [2]: Great announcements ...
"It is shown that if the Loren(t)z condition is discarded, the Maxwell–Heaviside field equations become the Lehnert equations, indicating the presence of charge density and current density in the vacuum. The Lehnert equations are a subset of the O(3) Yang–Mills field equations. Charge and current density in the vacuum are defined straightforwardly in terms of the vector potential and scalar potential, and are conceptually similar to Maxwell's displacement current, which also occurs in the classical vacuum. A demonstration is made of the existence of a time dependent classical vacuum polarization which appears if the Loren(t)z condition is discarded. Vacuum charge and current appear phenomenologically in the Lehnert equations but fundamentally in the O(3) Yang–Mills theory of classical electrodynamics. The latter also allows for the possibility of the existence of vacuum topological magnetic charge density and topological magnetic current density. Both O(3) and Lehnert equations are superior to the Maxwell–Heaviside equations in being able to describe phenomena not amenable to the latter. In theory, devices can be made to extract the energy associated with vacuum charge and current."

Sketch of the authors' erroneous idea

In [1] B. Lehnert and S. Roy have recalled to their readers that the Maxwell equations can be solved by means of potentials A, Φ, which are solutions of the differential equations

[R;(7a)]                                 1/ Att− ΔA + Ñ (div A + 1/ Φt) = μo j


[R;(7b)]                                 1/ Φtt − ΔΦ − (div A + 1/ Φt) t = ρ/εo .

without any coupling between A and Φ to obtain the solutions B = curl A and E = − ÑΦ − At. However, in addition the well-known Lorenz condition

div A + 1/ Φt = 0

can be imposed on the potentials to reduce the equations [R;(7a,b)] to inhomogeneous wave equations the solutions of which have well-known integral representations.

However, due to unclear formulations in [1] the authors believe the equations [R;(7a,b)] with applied Lorenz condition, the equations

[R;(7a')]                                 1/ Att− ΔA = μo j


[R;(7b')]                                 1/ Φtt − ΔΦ = ρ/εo .

to be the "Lorenz-free" equations to be used. And so the forgotten Lorenz term gives rise to a "vacuum current density" just of the size of the forgotten term:

jA = − 1/μo [grad (div A + 1/ Φt)] .

And none of the 15 authors became suspicious.

The authors consider the Maxwell equations in vacuo. We have

curl E + Bt = 0                                                                 (7)

curl B1/Et = 0                                                                 (8)

As outlined in the preceeding review we obtain for the vacuum case due to ρ = 0 and j = 0

[R; (7a)]         A + grad (div A + 1/ Φt) = Att − ΔA + grad (div A + 1/ Φt) = 0

which can be resolved for A to obtain

A = − grad (div A + 1/ Φt) =: μo jA                                                 (10)

The authors consider the field jA as "vacuum current density".

Now they try to go the way back to the Maxwell equations by means of the equations

B = Ñ×A,                 E = − AtÑΦ = 0                                                 (9)

The equation B = Ñ×A yields trivially div B = 0. Application of Ñ× to E + At = grad Φ yields the Faraday equation (7), Ñ× E + Bt = 0 .

The application of Ñ· to the second equ.(9) yields immediately

Ñ·E = − Ñ·At + ΔΦ = Φ − (Ñ·A + 1/ Φt )t = 0

due to

[R;(7b)]                                 1/ Φtt − ΔΦ − (1/ Φt + div A)t = ρ/εo         for ρ = 0 .

However, since the use of the Lorenz condition is forbidden, not by means of [1; (8)].

Analogously the use of [R;(7a)] with j = 0 leads back to the homogeneous Ampère equation (8).

Now the authors reconsider the equation (10) "with new eyes": The co-authors B. Lehnert and S. Roy assure erroneously that their equation [1; (7)] (instead of [R;(7a)]) is the "Lorenz-free" version (i.e. without assuming the Lorenz condition) - a simple mix-up, the contrary is true as was shown in the preceeding review. So being gullible the whole authorship unisono concludes that from (10) one can deduce that

curl B1/Et = jA                                                                 (12/13)

Once more: The correct equation to be used in the "Lorenz-free" case is [R;(7a)], not - as the authors falsly assume - the Equ. [1;(7)]. So (10) must be rewritten in the form of [R;(7a)] as

[R;(7a')]                 1/ Att− ΔA + Ñ (div A + 1/ Φt) = μo j := μo jA + Ñ (div A + 1/ Φt) .

This leads to

curl B1/Et = μo j = μo jA + Ñ (div A + 1/ Φt) .                                

However, due to (10) the right hand side is vanishing, i.e. instead of the wrong equation (13) due to

μo jA + Ñ (div A + 1/ Φt) = − Ñ (div A + 1/ Φt) + Ñ (div A + 1/ Φt) = 0 .

we finally obtain

curl B1/Et = 0 .                                                 (13')

The use of the correct "Lorenz-free" Lehnert-Roy equation [R;(7a)] instead of
the "Loren(t)z"-equation [1;(7)] yields a ZERO current in the Ampère equation (7).

The same turns out for the Gauss equation [1;(4)], where, using correctly [R;(7b)] instead of [1;(7)], no vacuum density ρA will appear.

Therefore there is no reason to speculate about vacuum current and vacuum energy and even applying the O(3) Yang-Mills Theory as one of the authors (M.W. Evans?) proposed.

Besides: M.W. Evans' O(3) theory is not Lorentz invariant and thus is no physical theory [5].

Both vacuum current and vacuum energy here had their very trivial origin in a mere mix-up of two equations in [1].


[1]   B. Lehnert and S. Roy; Extended Electromagnetic Theory ... ",
        APEIRON Vol.4, Nr. 2-3 1997

[2]   B. Lehnert, S. Roy e.a.; Classical Electrodynamics without the Lorentz Condition:
        Extracting Energy from the Vacuum
, Physica Scripta Vol. 61, No. 5, pp. 513-517, 2000,

[3]   B. Lehnert, S. Roy, An Extended Electromagnetic Theory,
        APEIRON Vol.7, Nr.1-2, January-April 2000

[4]   Alex Kaivarainen, Bo Lehnert, Two Extended New Approaches to Vacuum, Matter & Fields
        arXiv:physics/0112027v7 (2005)
        Abstract         pdf full text

[5]   G.W. Bruhn, On the Lorentz Behavior of M.W. Evans' O(3)-Symmetry Law,

[6]   G.W. Bruhn, Gauge Theory of the Maxwell Equations