Re: field transformations

[Joe Heafner <heafnerj@VNET.NET>]

On Thursday, Apr 3, 2003, at 14:22 US/Eastern, John S. Denker wrote:

> Please allow me to nit-pick the term "form

Nit-pick away.

> invariance".  The point is that the expression
>   E = -(1/c)dB/dt
> is not invariant as to form when we change from
> one frame to another.  Presumably everybody knows
> this;  it is the topic of this thread.

del x E = (-1/c)dB/dt is what I typed. I guess the del got lost since I
typed it as "\/".

Anyway, I know that E and B have different components in different
frames.

> To say the same thing in other words:  The physics
> described by the Maxwell equations _collectively_ is
> the same in the new frame, but if you consider one
> Maxwell equation separately it will go haywire.

Maybe, but so far everything works out nicely except the transformation
of the x component of the curl of E.

> E and B are the components (in a given frame) of a higher-
> grade object F, the electromagnetic field.  Suppose Bob
> is moving relative to Alice.  Then a field F that is purely
> electrical according to Alice will be partly magnetic
> according to Bob.  The real physics (F) will be the same,
> but the decomposition into components (E and B) will be
> different.

Yes, I completely understand this.

> This is entirely analogous to the behavior of Cartesian
> coordinates.  Suppose Doug's frame is rotated relative
> to Carol's.  Then they will disagree as to the x-coordinate
> and y-coordinate of any given object.  The physics will
> be the same, but the decomposition into components will
> be different.

Again, I understand this.

> Something is "invariant" if it doesn't change.  We don't
> expect components to be invariant.  They might be covariant
> or contravariant, but not invariant.

Again, I understand this.

> Every bivector can be represented as a tensor, so if
> you understand tensors you get bivectors for free.

But I don't understand tensors completely yet because no one has ever
been able to explain them to me. Most of the text books aren't helping
either, just spouting definitions in terms of transformations and
all...very confusing to me.  I don't really feel too badly about it
thought since many practicing physicists (including one author of a
widely read relativity text) tell me they don't understand tensors
either. :-)

> The tensors that correspond to bivectors are by far
> the easiest tensors to visualize.  Just as you can represent
> a vector using a stick (with a direction marked on it),
> you can represent a bivector using a piece of cardboard
> (with a direction of circulation marked on it).  You
> can make models and wave them around.

Understood.

> > What I've been looking for is a way to derive this transformation
> > without using tensors or four vectors.
>
> I still don't understand why this would be considered
> desirable.

Mere curiosity on my part, just to see WHETHER or not it can be done.
Eyges explicity demonstrates it, but not for all components of del x E.
I'm only trying to complete his own derivation.

> I vividly remember my first physics TA saying "our goal
> is not to teach you to do Lorentz transformations; the
> goal is to _avoid_ doing Lorentz transformations."  The
> point is that more-or-less any real-world problem worth
> doing can be done most easily using four-vectors.  Were
> not talking about elegance for the sake of elegance, or
> elegance at the expense of simplicity.  Were talking
> about just plain easier.  Better results with less effort.

Well, I could just as well ask "Why _avoid_ doing Lorentz
transformations?"

I agree with the last sentiment, BUT in an introductory course no one
knows about four vectors. Lorentz transformations of momentum and
energy and space and time can be done without explicit use of four
vectors so why not fields too?

> To me, Lorentz transformations are just mathematics,
> whereas four-vectors are real physics.  I can visualize
> four-vectors.  I can make models.  I can make diagrams.

I thought Lorentz transformations *define* four vectors, at least their
transformation properties anyway. Perhaps I was wrong.

> > 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.
>
> I worry that such an approach might tend to promote
> rote learning without imparting any real feel for
> the physics.

Rote learning is expressly forbidden in my classes. :-)

Geez, another exercise in futility. This is why I hesitate to even post
any more.

Joe...
... who turns around and goes back into his office, closing the door
behind him.