Evans' Basics of the Lorentz Transform

Gerhard W. Bruhn, Darmstadt University of Technology

Sept 15, 2008
Quotations from Evans' writings in

Subject: 119(1): Basics of the Lorentz transform Date: Mon, 15 Sep 2008 07:51:18 EDT

These are the basics of the famous Lorentz transform, which was worked out by Lorentz after correspondence with my predecessor Heaviside and with Fitzgerald. It can be seen that there are two Lorentz transforms, the boost and the rotation. The one relevant to the B cyclic theorem is the rotation, under which

i x j = k

is an O(3) symmetry rotation generator relation. If a vector is rotated, this relation is still the same, so is Lorentz invariant. Therefore the B Cyclic Theorem is invariant under Lorentz rotation, because the B Cyclic theorem is a rotation generator relation, QED. The Lorentz group consists of three boost generators and three rotation generators. The Poincare group consists of these and four more spacetime translation generators. The editor of Foundations of Physics, G. ‘t Hooft, is not aware of the basics of the subject, because he published a pseudo-paper asserting that i x j = k is “not covariant”. This is complete nonsense. In the Lorentz rotation (Carroll eq. (1.17)) the time component is unchanged, so we may consider the Cartesian frame with unit vectors i, j and k. If this frame is rotated, then we still have a Cartesian frame, and so i x j = k is still true. The only way is which a Lorentz boost can be applied to rotation is to use v = 0, because the rotation generators are independent of the boost generators. This is well known, and this note is meant as an educational background note.

Attachment: a119thpapernotes1.pdf

The result of this ''educational background note'' is really remarkable. We read on p.5:

                eo' = cosh Φ eo − sinh Φ k                                                 (35)

In frame K:

                k = i × j                                                                                 (36)

which is the usual Cartesian frame definition. Under the Lorentz boost in Z:

                ii,     jj,     k → − eo sinh φ + k cosh Φ ,                 (37)

OK so far. But there is a problem: What does this vector eo do here?

Evans' solution:


                cosh Φ = 1,     sinh Φ = 0                                                 (38)


                v = 0                                                                                 (39)


The relation (36) is an O(3) rotation generator relation which remains constant under the Lorentz boost. In the rotational Lorentz boost (Carroll (1.17)

                . . . . . . . . .     ,                                                                 (40)

and time is unchanged. We may therefore consider only the space relations (36).

This is Evans interpretation of a clearly written introductory text on the Lorentz transform on p.4 ff. in S.M. Carroll's Lecture Notes on GR [1]. Evans concludes that the relative velocity between the involved inertial frames must vanish - which means Lorentz transform only between frames with relative velocity v = 0. A really overwhelming method of reducing SRT to triviality.

This ''conclusion'' has a predecessor in Evans' book [2, p.89] where he is confronted with the problem that his B cyclic symmetry would require

                B(o)' = (1−v/c / 1+v/c)½ B(o)                                 [2, Eq.(3.111)]

while - as is well known [3, Eq.(11.148)] - the boost component of the magnetic flux does not change, B(o) = B(o)'. Evans solves this problem as surprising as wrong:

The root of this paradox is found in the fact that the plane wave is already propagating at c in frame K', and in consequence frame K' cannot move with respect to K at any velocity . . . The only possible solution of the paradox is v=0 in Eq.(3.111), leading to B(o) = B(o)'.

By the way: This was the first refutation of Evans' B Cyclic theorem by using the Lorentz transform (and by M.W. Evans himself!) before other guys gave refutations as well.

Subject: Proceeding with Paper 119 Date: Wed, 17 Sep 2008 06:06:20 EDT

In note 119(2) I intend to provide another educational background note on the invariance of i x j = k under the rotational Lorentz transform. This should be quite clear to a first year undergraduate but I give the proof to counteract the deliberate misinformation published by ‘t Hooft, who published a paper in “Foundations of Physics” asserting that i x j = k (B Cyclic Theorem) is somehow “not covariant”. Thsi si why I ask for ‘t Hooft’s resignation, and non-submission of papers to FP until another editor takes over who is not associated with Bruhn. Preferably van der Merwe should bd re-instated. The B Cyclic Theorem is the basic rotation generator relation itself, and time does not enter into it (see Carroll chapter one). Then in following notes I will go on to the main theme of paper 119, which is development of gravitomagnetism, dipole dipole interaction and so on. Paper 119 will also give a plausible first approximation for the equinoctial precession and hopefully will be finished before the September equinox.


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

[2] M.W. Evans e.a., The Enigmatic Photon - New Directions, Vol.4, Kluwer 1998, ISBN 0-7923-4826-5

[3] J.D. Jackson, Classical Electrodynamics, John Wiley & Sons 1999, ISBN 0-471-30932-X

Continuation here