## A Remark on V.V. Dvoeglazov's Lorentz Transform of the Electromagnetic Field and M.W. Evans' O(3)-Hypothesis

### Gerhard W. Bruhn, Darmstadt University of Technology

In the following text quotations from the paper [1] are displayed in black with equation labels (nn) at the right margin.

In [1; (11b)] V.V. Dvoeglazov displays a transformation formula for the longitudinal component B(3) of Evans' hypothetical B(3)-field:

B(3)' = B(3) + γ−1/β² (β·B(3))β − γβB(0)                                                 (11b)

where β = v/c, β = |β| and γ = 1/sqr(1−β²). For sake of simplicity we may assume c=1.

In the special case of longitudinal motion v = v k, v > 0, using B(3) = B(0)k Dvoeglazov's equation yields

B(3)' = B(0) sqr(1−β/1+β) k ,

which means B(3)' = B(3) sqr(1−β/1+β) , i.e. due to Dvoeglazov the longitudinal component of the B-field is decreased by the longitudinal Lorentz transform.

Since B(3) is merely the longitudinal component of a field B that has an additional transversal component B^ Dvoeglazov's result contradicts the well-known Lorentz transform of the electromagnetic field where the longitudinal component remains unchanged. We quote here the transformation rules from the "classical" book of A. Sommerfeld [2; p.245-246]

E| |' = E| | ,                                 B| |' = B| |                                 (| | = longitudinally),
E^' = 1/γ (E + v×B)^ ,                 B^' = 1/γ (Bv×E)^                 (^ = transversally).

In Equ.(6) Dvoeglazov shows us the transversal B-field components

Bx = − sqr(2) B(0) sin Φ ,                 By = + sqr(2) B(0) cos Φ .                                 (6)

which yields the transversal B-field

B^ = Bx i + By j = 1/sqr(2) B(0) [(i i + j)eiΦ + (− i i + j)eiΦ]                                 cf. (2)

where Φ = ω(t−z) for a wave propagating in z-direction with velocity c=1. The Maxwell equation Ñ ×B^ = E^/∂t leads to a corresponding transversal E-field

E^ = 1/sqr(2) B(0) [(ii j)eiΦ + (i + i j)eiΦ] = Ex i + Ey j .

Together with the longitudinal B-component

B| | = B(3) = B(0) k                                                                 (3)

we have all required field components to perform the above mentioned Lorentz transformation of B to obtain (note β=v due to c=1):

B| |' = B| |         and         B^' = 1/γ (B^ − β k×E^) = sqr(1−β/1+β) B^

### This result proves that V.V. Dvoeglazov's Equ. (11b) cannot be true.

More, we obtain an important conclusion concerning M.W. Evans O(3)-hypothesis [4; Chap.1.2]: Dvoeglazov's wave (2-3) fulfils Evans' Symmetry Relation relative to the original coordinate frame (x,y,z,t):

B(1) × B(2) = iB(0) B(3)* ,                                                                 (1)

which is obviously equivalent to

½ |B^|² = ½ (|Bx|² + By|²) = Bz|² = |B| |

However, seen from the new coordinate frame (x',y',z',t') after the longitudinal Lorentz transform we obtain

½ |B^'|² = ½ (|Bx'|² + By'|²) = 1−β/1+β |Bz|² = 1−β/1+β |B| ||² = 1−β/1+β |B| |'|²   <   |B| |'|² .

### References

[1]    V.V. Dvoeglazov, Comment on the 'Comment on the Longitudinal Magnetic Field ...’
by E. Comay ...
,
http://132.236.180.11/pdf/physics/9801024

[2]    A. Sommerfeld, ELEKTRODYNAMIK,
Akademische Verlagsgesellschaft Geest & Portig, Leipzig 1949

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