Hurray! I think I finally understand the problem in this derivation. I
have recently reviewed the applicable sections in Einstein's
"Relativity: The special and general theory", and I now feel better able
to make constructive comments in this thread.
In a former message,
> Subject: Einstein's third axiom (was: ...affirming the consequent)
> Date: Tue, 15 Apr 2003 10:15:41 +0300
Pentcho Valev listed Einstein's axioms as:
> 1. Principle of special relativity
>
> 2. Postulate of the constancy of the speed of light: If and only if the
speed of
> light in the first inertial frame is c >> v, then in another inertial
frame having
> a speed v with respect to the first it is c as well.
>
> 3. Postulate of the variability of the speed of light: If and only if
the speed of
> light in the first inertial frame is as low as v (x = vt), then it is
zero in the
> other inertial frame (x' = 0).
Now in my classes I say that Einstein's postulates were:
1. Principle of special relativity: all the laws of physics have the
same form for observers in all reference frames (observers that are not
accelerating or experiencing gravitational forces). [This means that if
I see you moving past me at a constant velocity (while at the same time
you see me moving past you at an equal speed in the opposite direction),
there is no physical experiment either of us can perform to determine
which of us is "really moving" or "really at rest". In fact, those
expressions no longer have any meaning, since any inertial frame is
equally good. We'll measure different values of position, speed, force,
electromagnetic fields, etc., but the equations that relate these
variables will be the same, no matter which reference frame we use.
There is a rich literature tracing the roots of this principle back to
Galileo and Newton, giving a reason for the idea of inertia, etc. The
fact that the laws of physics aren't different in the summer than in the
winter (the Earth is travelling in the other direction in its orbit
around the Sun, but at any moment its velocity is nearly constant, thus
providing a nearly inertial frame of reference) is a good start toward a
gut acceptance that "uniform motion" is not intrinsically different from
being "at rest": it's all relative to what frame of reference we use --
what is the uniform motion or rest in respect to?]
2. Postulate of the constancy of the speed of light: All inertial
observers measure the speed of light in vacuum as the same constant
value, c.
Clearly, Pentcho Valev's version of the 2nd principle had extra
qualifiers which I couldn't accept, and at the time I was very troubled
particularly the 3rd principle, which at first glance didn't seem like
anything I'd ever read before. I tried to create an equivalent version
which might help clarify whether it was really what Einstein was
saying. I was not satisfied with either version, but had no more time
to pursue the matter, so I let it drop until now.
Then, in a later message,
> Subject: Relativity and the premise x'=0 <-> x=vt
> Date: Wed, 07 May 2003 09:09:53 +0200
Pentcho Valev rephrased and renumbered the axioms as:
> Premise 1: x' = ax + by + cz + dt
>
> Premise 2: x' = 0 <-> x = vt
>
> Premise 3: x = 0 <-> x' = -vt'
>
> where A <-> B means "If and only if A then B". Premises 1-3 yield the
> following results:
>
> t' = at ; d = -va
...
I did not read this message at the time. I wish I had! I now see what
was wrong with the version of principle 3 given in the earlier thread.
More important, I now recognize it for what it really is -- not a
"postulate of the variability of the speed of light", but a combination
of the worldline description of one observer in the reference frame of
the other, and the synchronization condition. (Details below.)
Finally, in a recent message, Pentcho Valev restates the above steps,
providing comments. I will now comment on the this:
> Subject: Re: Teaching logic is urgent (the only reasonable
transformations)
> Date: Thu, 05 Jun 2003 14:04:40 +0200
> From: Pentcho Valev <pvalev@BAS.BG>
>
> Pentcho Valev wrote:
>
> > --- 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/
This is, of course, the condition that the primed observer is moving at
speed v with respect to the unprimed observer, and the two observers
coincide momentarily at time t=0. x'=0 describes the worldline of the
primed observer in her own frame: she remains at the origin of the
primed frame for all time t', so x'=0 no matter what t' is. x = vt
describes the same worldline of the primed observer in the unprimed
frame: her x-position moves in the +x-direction as time (t) passes, at
a constant speed v. Letting her position be at the origin when t=0 we
have x = vt.
> > which, combined with /1/, gives
> >
> > d = -av /3/
> >
> > The condition symmetrical to /2/ is
> >
> > x = 0 <-> x' = -vt' /4/
x=0 describes the worldline of the unprimed observer, who remains at the
origin of the unprimed frame. This symmetrical condition also specifies
that the unprimed observer is moving at speed -v with respect to the
primed observer, and the two observers coincide momentarily at time
t'=0. These conditions are consistent with those inherent in /2/ iff
the two observers' clocks are synchronized (t=t'=0) at the moment they
coincide. This is the standard assumption, and it's fair to use it
"without loss of generality", since we could always go back and add
offsets to one or both of the times t and t' if desired.
> > 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'.
It is fundamental in logic to distinguish between the truth of a
compound statement such as p <-> q and the truth or falsehood of the
individual statement p and the individual statement q. /2/ makes the
claim "p <-> q" (here: x' = 0 <-> x = vt). Even if we accept the
truth "p <-> q" we must be wary of then making the mistake of believing
that either "p" or "q" is necessarily true. "p <-> q" only guarantees
the sameness of the truth-values of p and q: they are either both true
OR both false. 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.
In short,
a. /2/ is valid iff the relative speed of observer O' with respect to
observer O is v and O & O' coincide when t=0.
b. /4/ is valid iff the relative speed of observer O with respect to
observer O' is -v and O & O' coincide when t'=0.
c. /2/ and /4/ are both valid iff the relative speed of observer O' with
respect to observer O is v and O & O' coincide when t=t'=0.
HOWEVER, this only relates to the validity of the "<->" (iff, if and
only if). It does not allow one to apply both /2/ and /4/ to any
arbitrary point in spacetime! Remember what they mean: /2/ identifies
the worldline of O', /4/ the worldline of O. If we insist that both /2/
and /4/ be true for some point, that point must be the intersection of
the two worldlines. Mathematically we can verify this:
x=0 and x'=0
<-> (x=vt) and (x'=-vt')
<-> (x=0 and x=vt) and (x'=0 and x'=-vt')
<-> (x=0 and 0=vt) and (x'=0 and 0=-vt')
<-> (x=0 and t=0) and (x'=0 and t'=0).
Thus /2/ and /4/ are both true only for the origin. [The implications
work in both directions if /2/ and /4/ are true, and if v is not equal
to zero, i.e., if our two observers are in uniform motion with respect
to one another.]
> This forces us to advance THE ONLY REASONABLE TRANSFORMATIONS compatible
> with /1/, /2/ and /4/. These are
>
> x' = a(x - vt) /6/
>
> t' = at /7/
>
> Obviously the transformations /6/ and /7/ are incompatible with the
> postulate of the constancy of the speed of light (x=ct <-> x'=ct'). As
> for the coefficient a, it could be determined as follows. If time
> dilation is experimentally proved and processes in the moving system are
> slower than in the system at rest, we can substitute, in /7/, a = 1/gamma
> and then
>
> x' = (1/gamma)(x - vt) /8/
>
> t' = (1/gamma)t /9/
>
> If time dilation is an unreliable experimental finding, we substitute a =
> 1 and return to Galilean transformations.
>
> Pentcho
Now it can be seen that the individual parts of /2/ and /4/ must not be
applied together to points other than the origin of the two reference
frames, and indeed most of spacetime is neither on the worldline of O
nor on that of O', so the individual parts of /2/ and /4/ are not in
general true. Our transformations need only allow them to be true as
(separate) special cases, and of course, this is what the Lorentz
transformations do. [* see follow-up message] Thus the derivation of
/6/-/9/ above is
flawed. Moreover, as Pentcho himself points out, they are incompatible
with the postulate of the constancy of the speed of light. This
principle has been repeatedly tested and found reliable, from the
Michelson-Morley experiments on to more modern tests. In itself, this
eliminates any transformation that does not agree with the constancy of
the speed of light (at least within the limits of experimental error).
In fact, as has been previously mentioned, the Lorentz transformation is
the only consistent linear transformation equation that can be derived
from Einstein's postulates.
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.