Re: Some Beginner SR Questions
- From: "Dean Elliot" <anonymous@xxxxxxxxxx>
- Date: Fri, 13 May 2005 02:41:08 GMT
"Bill Hobba" <bhobba@xxxxxxxxxxxxxx> wrote in message
news:7Km_d.1819$C7.1305@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> "Dean Elliot" <anonymous@xxxxxxxxx> wrote in message
> news:9om_d.24015$OU1.17599@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> > In
> > "ON THE ELECTRODYNAMICS"
> > OF MOVING BODIES
> > By A. Einstein
> > June 30, 1905
> > at http://www.fourmilab.ch/etexts/einstein/specrel/www/ (and
> > http://www.wiley-vch.de/berlin/journals/adp/890_921.pdf )
>
> If you are a beginning SR student then Einstein's 1905 paper is
probably not
> the best place to start since things have moved on considerably since
then.
> Have a look a the following links then come back to it:
> http://arxiv.org/abs/physics/0110076,
> and ancient, but I still think excellent post by Tom Roberts
>
http://groups.google.com/groups?hl=en&lr=&c2coff=1&selm=54jfst%24glp%40s
sbunews.ih.lucent.com
> and chapter 10 of
> http://www.courses.fas.harvard.edu/~phys16/Textbook/
> under the heading of Relativity without c.
>
> Thanks
> Bill
>
Thank you very much. I've found all of them interesting. I have
different issues with them, reflecting my difficulty in understanding
them, but commonly upon the explanation of the linearity of the
coordinate transformations. I would appreciate clarrifications of the
issues that cause my misunderstandings even though they seem obvious to
you.
I have a number of issues with Tom Roberts derivation mostly to do with
the style.
Here is a broad sampling:
In his justification of the linearity of the transformations, he states
that "The Mapping Postulate and the Homogeneity postulate imply that the
transformation equations are linear.." which is not convincingly
apparent to me (though the linearity of the transforms "seems" to be
consistant with the Homogeneity Postulate). He seems to justify this by
saying "... if there were other powers of x or t on the right-hand side,
the transformation would not be one-to-one everywhere." when it seems to
me that if there were odd powers of x or t then the
transformation could still be one-to-one.
Other examples include:
He derives the identity C(u) = A (u) using the special condition when v
= -u, and then states "... this is true in general (not just for v
= -u); it is a mathematical statement about the two functions, valid for
all u." which is a leap of faith that begs the questions as to why he
derived it only for the special condition.
When describing the Isotropy/Homogenietiy Postulate, he twice uses the
phrase "The transformation must have the same mathematical form...." It
is unclear to me what is meant by "the same mathematical form" from the
context in which they are made.
One issue that might be considered overly petty but that I still find
disturbing is when he describes the Mapping Postulate he says: "The
coordinate transformation from one system to the other MUST be
one-to-one and onto the other, BECAUSE THEY ARE DESCRIBING THE SAME
PHYSICAL SPACE-TIME; the transformation must be invertible (see
Relativity Postulate, below)." The capitalization is reminiscent of the
style of cranks when they try to force their opinion by intimidation
(really, there's no need to shout). If there needs to be a justification
for the postulate than I think it would have been better to say
something to the effect that it had to be one-to-one otherwise there
would be points in space in one frame would have several or undetermined
location in the other; and since the inverse has to be one-to-one also,
the transformation has to be onto as well. Or, as per Rostislav
Polishchuk's presentation, "Also it is clear that transformation
functions must lead to one-to-one transformations otherwise single
particle in one frame could appear as several (or have undetermined
position) in another. Hence transformation functions have an inverse."
I like Rostislav Polishchuk's derivation up through chapter IV. My only
issues are with some missing grammar (articles: "a" and "the")
throughtout which makes it sometimes more difficult to follow. I like
that he usually supports his conclusions with brief, informal proofs by
contradiction. I like his derivation of the linearity of the coordinate
transforms except that I find (erroneously of course) that his
explanation for the proportionallity of delta TAU to delta T is
circular.
I am flummoxed by several things in chapter IV including the sudden
introduction and brief use of the ox and ox' axes some of his language
and logic, for example the conclusion in the 7th paragraph "We have
shown that y' and z' do not depend on x and y,..." when he has only
shown that they do not depend on x and t, I think.
Chapter 10 of http://www.courses.fas.harvard.edu/~phys16/Textbook/ is
more my speed (I'm at least as much interested in Einstein's derivation
as in the theory itself), though I've barely glanced at it so far. But I
have problems with the explanation for the linearity of the transforms.
In section 10.3 (in section 10.8, it refers to an appendix I've yet to
read) it says "1. We have assumed in eq. (10.12) that delta-x and
delta-t are linear functions of delta-x' and delta-t'. And we have also
assumed that A, B, C, and D are constants (that is, dependent only on v,
and not on x,t,x',t').
The first of these assumptions is justified by the fact that any finite
interval can be built up from a series of many infinitesimal ones. But
for an infinitesimal interval, any terms such as, for example,
(delta-t')^2, are negligible compared to the linear terms. Therefore, if
we add up all the infinitesimal intervals to obtain a finite one, we
will be left with only the linear terms. Equivalently, it shouldn't
matter whether we make a measurement with, say, meter sticks or
half-meter sticks."
First, the equation referenced (10.12) is not expressed in terms of
infinitesimals.
Second, in order to build a finite interval from infinitesimal ones
requires an infinite series, in which case the non-linear terms are not
necessarily negligible. It seems as though what's being said is that the
tangent line to a curve at a point is a good approximation to the curve
at that point so one may ignore the curve's curvature in theory as well
as practise (i.e. globally as well as locally).
Third, non-linear terms can be of different forms than exponents of
terms (e.g. cos(t')).
In Dirk Van de Moortel's response, for brevity, he assumes that Tau
is linear in x' and t (though it seems as though it has to be since Tau
is expressed as a linear combination of x' and t and the diff eq that
defines Tau is expressed as a combination of dx' and dt without any
other expressions involving x' and t. That is, if Tau had non-linear
expressions of x' and t then the diff eq would have expressions
involving x' and t beyond dx' and dt.)
Thanks again,
Dean.
.
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