Re: Chubykalo and Vlaev's basic mistake

Tom Roberts wrote on Sun, 06 Jul 2008 22:32:55 -0500:

Juan R. González-Álvarez wrote:
Both you and Jackson are plain wrong.

Simply stating that is insufficient. You must DESCRIBE and EXPLAIN the
mistake.

You are not reading.

Merely repeating C&V's claims and mistakes is insufficient.

C&V did just what you are doing, and no more: they CLAIMED Jackson is
wrong, but did not really describe or explain his error.

This is also false. They pointed what was Jackson error. But I can see
below that you deleted their reply from my previous message.

You are deleting their explanation and then claiming there gave none!

(snip)

Here's more detail on this second error Jackson points out -- my
discussion uses the level of rigor common in physics (sufficient to
illustrate the error). C&V's response did not touch their actual error
-- it is rather ironic, because they forgot to include an IMPLICIT
dependency (which is the opposite of what triggered their entire paper).

They need to compute d|R|/dt0, and use this expression:

|R| = c(t - t0)

and conclude

d|R|/dt0 = -c (my ascii d is their \del)

This sure LOOKS like they simply did a partial derivative of that
equation holding t constant. What they actually did is more complicated
BUT IS EQUIVALENT (they are using a higher level of rigor than I, but
that did not prevent their basic error). The underlying problem is that
the notation does not include functional dependencies (C&V never mention
this), so let me put them in:

|R(x,y,z,t,path)| = c(t - t0(x,y,z,t,path))

That is, both R and t0 are completely determined by x,y,z,t and the path
of the source charge (remember x,y,z,t are the coordinates in this
inertial frame where the field is being measured).

Here is the conceptual source of your error.

The dependencies on
x,y,z and the path are irrelevant here, so let me hold them all constant
and omit them for clarity:

|R(t)| = c(t - t0(t))

Now one can invert t0(t) and use t0 as the independent variable, because
the trajectory is unique and |V|<c. So do so (formally):

|R(t(t0))| = c(t(t0) - t0)

One can of course trivially re-parameterize R to be written in terms of
t0:

|R(t0)| = c(t(t0) - t0)

Now one can compute d|R|/dt0 (both sides are purely functions of t0),
and their error becomes clear: they omitted the implicit dependence of t
on t0 (here made explicit).

In general one cannot invert t0(t) without knowing the trajectory of the
source charge (remember it's really t0(x,y,z,t,path)). But in a
neighborhood of t0(t), one can do so to first order (involving the
velocity V of the source), and since we only need the derivative of |R|
first-order is sufficient (all higher-order terms drop out of the
derivative). When you do this, you of course get the same answer Jackson
gave:

d|R|/dt0 = - R.V/|R|

You make the same mistake that Jackson did. Except Jackson did once and
was corrected in (physics/0205041v1) but you repeat the mistake after
being warned twice about it!

Once again, your (and Jackson)

d|R|/dt0 = - R.V/|R|

is wrong. The correct expression is

d|R|/dt0 = - c

Authors already warned in footnote c in page 3792 what *functional*
expressions yield the derivatives used. They repeated the warning in the
page 3 of physics/0205041v1 when they said (emphasis on the original)

(\blockquote
use the fact that two /different/ expressions of the function R exist:
)

And wrote again the *two* R functions, but now with more detail,
equations (8) and (9) on physics/0205041v1.

Authors also pointed to Ref 2 (Landau) in the footnote c of the original
paper for the *two* functional expressions.

(sniped)

The rest of this is just embroidery on things stated above (except for
your outrageous statement about Jackson's "ignorance", and your claims
about unpublished stuff invisible to me).

Jackson (hep-ph/0203076) looks perturbed by equation (3) because claims
is inconsistent with another equation in C&V (don't true) and want to
obtain an alternative expression via second derivative of R.

Then he computes (4) and from here he obtains expression (5). What is the
source of the mistake on Jackson?

He starts with

dR^2 / dt0 = 2 · R_j dR_j / dt0

but now Jackson confounds *numerical* equivalence with *functional*
equivalence [#]. For (8) and (9) in (physics/0205041v1) also page 3792
and footnote c on original C&V paper

R(t0) = R(t0) but dR/dt0 /= dR/dt0

Over that mistake Jackson extracts the time derivative from the *second*
function in C&V and uses as substitute for the time derivative of the
*first* function, obtaining the wrong final result (5)

dR / dt_0 = - ((\vect R) (\vect V)) / R

If in equation (4) you use the time derivative for the *first* R function
then you obtain the result

dR / dt_0 = - c

Jackson pointed out a third error completely disregarded in C&V's reply.
He points out an error in a previous paper that C&V used to justify
"instantaneous interactions". He calls their claims a "selective belief
in mathematics" (see hep-ph/0203076 for why). I merely point out that in
their paper C&V evaluate source properties ONLY at the retarded time, so
there is only a RETARDED interaction.

You may refer to section "5. Failure of Subsequent Arguments" in Jackson.
There Jackson continues showing a complete inability to understand the
arguments and mathematics on Ref. 2.

When Jackson says:

(\blockquote
On the basis of their wrong result, they argue for the necessity of both
conventional retarded interactions and instantaneous, action-at-a
distance interactions. Since the original demonstration is incorrect, the
subsequent arguments have no basis.
)

Jackson shows either he does not understand the paper or is using some
straw man argument.

Jackson further comments on gauges and mixed components also look
specially misleading.

(snip)

Noticing how the idea of action at a distance perturbs relativists is
funny. This reaction was also noticed by C&V:

(\blockquote
In Section 4 of our work^2 we showed that Faraday’s law is obeyed if
one considers the functions E and B as functions with both implicit
and explicit dependence on t (or on xi ).^a That is why we do not
understand why J.D.Jackson did the same in Section 4 of his work^1.
It seems to us that a basic reason behind Jackson’s antagonism to our
work is the following: our interpretation of the explicit time-dependence
as a certain manifestation of instantaneous action-at-a-distance and on
the other hand the implicit time-dependence (i.e. exclusively through
the relation (3)) as a well-known short-range action.
)

What will be relativists reaction when showed that one can prove that the
retarded LW potentials arise from a more fundamental instantaneous action
at a distance theory.

i)
The derivation of the LW potentials is rigorous (more rigor than in C&V
and Jackson papers for instance).

ii)
The paper shows what are the assumptions underlying the LW potentials
(and in all Maxwell electrodynamics and SR).

iii)
The new theory is free from inconsistencies characteristics of classical
electrodynamics and SR.

iv)
And has the bonus naturally explains all recent experimental results that
classical electrodynamics and SR failed to explain. Including well-known
experiments where the LW potentials give the wrong magnitude and even
sign!

In my work I also offer a microscopic explanation (in QED language) of
why the two R functions (8) and (9) are different. In QED, each function
R is related to a different quantum field.

Jackson and Tom Roberts confound both; C&V, Landau, and Lifshitz do not.


[#] For instance (y = x) and (y = x^2) numerically agree for (x = 1)
but derivatives (y' = 1) and (y' = 2x) do not.


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http://canonicalscience.org
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