Re: field transformations

[Joe Heafner <heafnerj@VNET.NET>]

On Friday, Mar 28, 2003, at 18:06 US/Eastern, John S. Denker wrote:

> I can't imagine deriving or even explaining the
> transformations without four-vectors.  For that
> matter I can't imagine introductory E&M without
> four-vectors.

Well, I meant "introductory" in the sense of the second semester of
that first introductory course (e.g. Halliday and Resnick). The text
I'm using, Matter & Interactions, presents a very nice discussion of
reference frames and fields. The authors specifically demonstrate how a
static E field in one inertial frame is seen as a combined E and B
field in another inertial frame. They do so using the Lorentz
transformation of E and B, but they do not present an actual derivation
of the transformation. They don't present a derivation of the standard
Lorentz transformation either.

What I've been looking for is a way to derive this transformation
without using tensors or four vectors. Today, I found a way on pp.
231-232 of Eyges' text _The Classical Electromagnetic Field_ (Dover,
1980). The method involves Lorentz transforming the components of \/ x
E = -(1/c)dB/dt and imposing form invariance. I'm in the process of
carrying out the whole derivation to make sure I understand it.

I agree what geometric methods are probably the most elegant, but I
don't understand them. I'm just now beginning to think I understand
tensors. Technically, I "learned" them in grad school from Goldstein
and Jackson, but in reality what I "learned" were mechanical
mathematical steps for working problems. The breakthrough for me
consisted of a thorough understanding of four vectors and a realization
that scalars, vectors, and four vectors are merely special cases of
tensors.

Cheers,
Joe Heafner