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Re: Teaching logic is urgent



--- Bob LaMontagne <rlamont@POSTOFFICE.PROVIDENCE.EDU> wrote:

At this very elementary level,
your statements (A), (B) and (C) appear logically compatible when
the
Lorentz transformations are used.

You proposed this as a question to be presented to students. Could
you
please present what you would consider an acceptable student
solution -
something more detailed than simple assertions that (A), (B) and
(C) are
incompatible.

This is the strangest problem in science I know of. Let us start with
the linearity condition

x' = ax + by + cz + dt

which is reduced to

x' = ax + dt /1/

Then Einstein introduces

x' = 0 <-> x = vt /2/

which, combined with /1/, gives

d = -av /3/

The condition symmetrical to /2/ is

x = 0 <-> x' = -vt' /4/

which, combined with /1/ and /3/ gives

t' = at /5/

The last result is obviously incompatible with the second Lorentz
equation, i.e. Lorentz transformations are incompatible with the
basic conditions /1/, /2/ and /4/ (since the incompatible result /5/
follows from them). I was right about that. However there is a trick
which I have never expected to see in science. The unfavorable
result /5/ must be changed, ILLEGALLY, in a way such that the
conditions /2/ and /4/ remained apparently fulfilled. So we
replace /5/ with

t' = at + mx /6/

where m is some coefficient to be determined later. The term mx is
illegal but this cannot be discovered by applying the conditions /2/
and /4/ - you said they are compatible with Lorentz transformations
and I agreed, but we were both misled. Since the term mx is illegal,
equations /5/ and /1/ are compatible but equations /6/ and /1/, which
are prototypes of the two Lorentz equations, are incompatible. Let us
now go directly to Lorentz equations and express x from the
counterparts of /1/ and /6/:

x = (tc^2 - t'*c^2*sqrt)/v (second Lorentz)

x = vt + x'*sqrt (first Lorentz)

Clearly in the second Lorentz x is almost inversely proportional to v
whereas in the first Lorentz x increases almost linearly with v
(the "almost" is due to the sqrt). The two Lorentz eqyuations are
incompatible.

Pentcho