Re: Too Trivial for Tom Roberts, Impossible for Most
From: Eugene Shubert (GalileoProject002_at_everythingimportant.org)
Date: 10/25/04
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Date: 24 Oct 2004 17:46:06 -0700
Tom Roberts <tjroberts@lucent.com> wrote in message news:<A9Hed.15741$5b1.13208@newssvr17.news.prodigy.com>...
> Eugene Shubert wrote:
> > http://www.everythingimportant.org/relativity/special.pdf
> The quality of a derivation of the equations of SR is usually judged by:
> 1. the simplicity and physical believability of its assumptions, or
> lack thereof.
> and:
> 2. the generality of those assumptions.
I wholeheartedly agree. And by the generality of those assumptions,
I assume you mean the value and added clarity that it brings to the
subject and to the opening of new questions, vistas and future
research. That describes my paper perfectly.
> Your derivation is seriously lacking in both of those aspects:
> 1. On page 2 your assumptions for \mu and \gamma are completely
> unsupported,
My official derivation begins on page 4. The assumption that you refer
to (page 2), the principle called change of variables, also called
substitution, is one of the clearest and most useful mathematical
ideas in high school algebra and baby calculus. What's your dispute
with intuitively simple fundamentals? What support do I need and why
is it an unwarranted physical assumption if I label the function
1/sqrt(1 -v^2/c^2) with the Greek letter gamma?
> and are not obvious at all -- these are QUITE
> unusual assumptions, to say the least; why should anybody
> believe them?
Why should anyone believe in the substitution v/c = tanh(theta),
which transforms the usual Lorentz transformation to its hyperbolic
form? It's just the writing of one parameter in terms of another.
Why is substitution allowable here but not there? Perhaps the
difference is that every child mathematician can plod through
a difficult mathematical proof, checking each step for logical
correctness, whereas mature physicists need their hands held by
someone they trust and have their irrational prejudices pacified
at every step.
> 2. Your sliding rulers work in 1 spatial dimension,
Thanks for mentioning that. You'd be surprised by the number of
physicists and seemingly educated folks who have an emotional
difficulty with sliding rulers!
> though you have left a lot out (e.g. how to mark them uniformly;
I suspect that just about all students of algebra at the high school
level will assume that the rulers are pre-made and that even middle
school students could figure out how "to mark them uniformly."
> how to know they move with uniform velocity);
They're assumed to move with uniform velocity. That translates into
equal distances traveled in equal proper times. That's where the
constant u comes from in the general two-ruler synchronization of
the Shubertian clock:
T = T(x,x') = -x'/u + xi(x)
T' = T'(x,x') = x/u + zeta(x')
> your omissions can be corrected. But it is not at all clear how to
> apply this to an arbitrary relative velocity in 3 spatial dimensions.
My intended audience is second year algebra students in high schools
and all backward uneducated folks on the newsgroups. Relativity in 3
spatial dimensions plus 1 time dimension is a college level topic.
> And you have completely left out any mention of isotropy and
> homogeneity, which are important and necessary aspects of inertial
> frames in SR.
Homogeneity and isotropy are geometric ideas of zero or virtually
zero importance in 1 spatial dimension.
> This is, of course, related to the omissions I mentioned
> in #2 above; but it is not obvious how to resolve this with your
> assumptions, especially isotropy.
I encourage students to try to break the no-nonlinearity postulate
of SR and construct all the Shubertian clocks possible that are
unauthorized and frowned upon in conventional physics. I advise that
students and researchers only do so for purely mathematical reasons
so that no sacred traditions are violated and that sacrilege not be
flaunted.
http://www.everythingimportant.org/viewtopic.php?t=221
http://www.everythingimportant.org/relativity/generalized.htm
> A better approach, IMHO, is to use group theory: given sufficient
> postulates to establish isotropy and homogeneity of the coordinates,
Homogeneity and isotropy are mathematical terms that describe a
geometry, not coordinates. You are perfectly free to break with
standard mathematical convention and define what you mean by
homogeneous and isotropic coordinates but I've never seen you do that.
I've seen you presupposing that space and time together is a geometry
called spacetime and that the homogeneity and isotropy of spacetime
automatically implies that coordinate transformations are linear.
The problem with your approach to SR is that you don't define what a
geometry is so it's impossible to really understand the implications
of homogeneity and isotropy in your vague, meaningless and nebulous
terms. I don't mean to discourage you from pursuing a geometric
derivation. It's just that the fallacy of supposed linearity of
coordinate transformations is easily refuted by simply defining
geometry according to Klein's Erlanger program:
"Every geometry is defined by a group of transformations, and
the goal of every geometry is to study invariants of this group."
-Klein, Erlanger Program.
"Each type of geometry is the study of the invariants of a group of
transformations; that is, the symmetry transformation of some chosen
space." Stewart and Golubitsky 1993, p. 44.
"A geometry is defined by a group of transformations, and investigates
everything that is invariant under the transformations of this given
group." Weyl 1952, p. 133.
The geometry of Minkowski space is defined by the Poincaré group.
http://en.wikipedia.org/wiki/Poincar%E9_group
Here's the critical point. It's easy for any child mathematician like
myself to show that the nonlinear transformation group of exercises 1
and 2 of http://www.everythingimportant.org/relativity/generalized.htm
is isomorphic to the Poincaré group. That means that their respective
geometries are isomorphic, i.e., indistinguishable. Thus, it's
impossible to prove linearity of coordinate transformations from
homogeneity and isotropy alone. If Minkowski space is isotropic and
homogeneous, then so is the geometry defined by my wildly nonlinear
transformation group.
> group theory constrains the transforms to 3 groups:
> The Euclid group in 4 dimensions
> The Galileo group in 3 dimensions
> The Lorentz group in (3+1) dimensions
> The first has grossly unphysical consequences, and the second does not
> agree with basic observations about the world, such as the simple fact
> that pion beams exist. But the Lorentz group works, and is the basis of SR.
>
> Evaluated on the basis of those above criteria, this approach is VASTLY
> simpler and more general than yours.
My approach is obviously more general and informative. The Shubertian
clock is the discovery I used to correctly understand and properly
interpret and the first counterexample to Einstein's misunderstood and
thoroughly misguided no-nonlinearity postulate of special relativity.
That's a new result.
When you remove the nonsense argument about homogeneity and isotropy
implies linearity, your approach will be simple. But I use group
theory also and you should notice that I begin with the greatest
conceivable nonlinearity possible and then I quickly and honestly
simplify the problem to linear mathematics, two unknown functions and
an easily invertible matrix. What I've done is spend a lot of time
explaining an intuitively simple and straightforward definition of
time--the Shubertian clock. That's to my credit.
> It was already old when I first saw it ~1972.
What you have is just a slight rewrite of the papers of Ignatowsky,
Frank and Rothe in papers written between 1910 and 1912.
http://groups.google.com/groups?selm=CQ1R9.535927$%m4.152001@rwcrnsc52.ops.asp.att.net
> [I posted a version of this some 15-20 years ago, but realize
> my ancient presentation has some major flaws (which can be
> corrected).]
> Tom Roberts tjroberts@lucent.com
Eugene Shubert
http://www.everythingimportant.org/relativity/special.pdf
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