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Re: Teaching logic is urgent (the only reasonable transformations)

Bob, you are right that /5/ below applies to those events that occur at x=0 and in
this sense my argument is inconclusive, but this still does not mean that the
x-containing term in Lorentz second equation is legitimate. Let us first see what
"event" means. There is at least one case in special relativity in which "event" is
reliably defined. For the front of a beam starting at the origin (x=x'=t=t'=0) and
moving along the x-axis Einstein postulates (x=ct <-> x'=ct'), i.e. in this case the
event is the movement of the front of a beam. To avoid confusion, further I shall
deal with the movement of the front of a beam only.
For a beam starting at the origin and moving along the y'-axis (on the train)
we obviously have x' = 0. For x' = 0, Einstein postulates x = vt (in the track
frame):

x' = 0 <-> x = vt (/2/ below)

Let us combine this with the system

x' = ax + dt (/1/ below)

t' = px + qt

So we obtain

d = -av (/3/ below)

t' = (pv + q)t

The above equation is again incompatible with Lorentz second equation but this time
the incompatibility is fatal - the coefficient (pv + q) is independent of x whereas
in Lorentz second equation there is a x-dependent term. And the above equation
characterizes an event that DOES NOT occur at x = 0.
My claim that

x' = a(x -vt)

t' = at

are the only reasonable transformations remains.

Pentcho

Bob Sciamanda wrote:

Ken's analysis is fine except for his statement that Pentcho's
| "equations /3/ and /5/ above likewise only hold true for points on the
worldline of the primed
| observer, i.e., for points where x'=0."

Pentcho's (3) is a statement involving only constants and is true in general.
Statement (5) is true only for events occuring at x=0 (the unprimed observer's
position).

As I earlier observed in a private email to Pentcho, his treatment from eq (1)
to eq (5) is correct. It is his interpretation of (5) which is in error. As Ken
has also written, (5) does not apply to all events; it relates the (t and t' )
time coordinates only of THOSE EVENTS WHICH OCCUR AT X = 0.

Re Ken's side note:
It was in fact the invariance of the speed of e/m waves as a special case of the
relativity principle that motivated Einstein. Maxwell's equations predict a
definite propagation speed for e/m waves in vacuum without reference to any
particular inertial frame (the same is true of the particle velocities occuring
in Maxwellian magnetic interactions). Compare this circumstance with the
derivation of the acoustic wave equation which obviously derives a propagation
speed relative to the acoustic medium. (The only speculative way out is the
invention of an intangible medium - the ether. Michaelson-Morley et al kills
that!)

Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor
----- Original Message -----
From: "Ken Caviness" <caviness@SOUTHERN.EDU>
To: <PHYS-L@lists.nau.edu>
Sent: Thursday, June 05, 2003 4:54 PM
Subject: Re: Teaching logic is urgent (the only reasonable transformations)

| > > 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/

SNIP
| > > which, combined with /1/, gives
| > >
| > > d = -av /3/
| > >
| > > The condition symmetrical to /2/ is
| > >
| > > x = 0 <-> x' = -vt' /4/
|
SNIP
| > > 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/.
|
| There is indeed an incompatibility shown here, but it is the
| incompatibility of assuming both that x = vt and that x' = vt'.
SNIP
| Note that we wish to find constants a and d such that
| /1/ can be generally applicable for all points describable in the two
| reference frames, by (x,t) and (x',t'), respectively. But /2/
| distinguishes points on the x' axis, the worldline of the primed
| observer, that is to say, both parts of the "if and only if" are true
| for such points, and false elsewhere. The statement "q" (here: x = vt)
| _cannot_ be applied to all points. Your equations /3/ and /5/ above
| likewise only hold true for points on the worldline of the primed
| observer, i.e., for points where x'=0.
SNIP
| As a side note, it is quite interesting to consider the constancy of the
| speed of light as a special case of the first principle of special
| relativity, that of the indistinguishability of reference frames. If
| light did travel with different speeds according to different inertial
| frames, an experiment could reveal the existence or absence of absolute
| motion, and the equations of physics would have a unique, simple form in
| the one frame at "absolute rest". Thus the second principle can be
| viewed as a corollary of the first.
| Fun! Ken