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Re: Einstein's third axiom (was: ...affirming the consequent)

I have always thought that the linearity assumption is a case of Occam's
Razor.

Joel Rauber

-----Original Message-----
From: Ken Caviness [mailto:caviness@SOUTHERN.EDU]
Sent: Tuesday, April 15, 2003 1:36 PM
To: PHYS-L@lists.nau.edu
Subject: Re: Einstein's third axiom (was: ...affirming the consequent)


Pentcho Valev wrote:

Ken Caviness wrote:

Pentcho Valev wrote:

However Einstein needs something different:

B -> A, A, therefore B /4/

He proceeds in accordance with /4/ - builds Lorentz
equations on B and
so creates the sequence (A therefore B therefore
Lorentz equations) -
the illusion is that
Lorentz equations ultimately stem from A. In fact, A
CAN be a corollary
of B or Lorentz equations, in accordance with /3/, but
Lorentz equations
can BY NO MEANS be
deduced from A.

I vividly remember an assignment in my freshman year
Engineering Physics class
where we were asked to _derive_ the Lorentz
transformation equations from
Einstein's postulates that the laws of physics and the
speed of light are the
same for all inertial observers. I have since used this
in my classes or as a
homework assignment. The only additional assumption
needed is that the
transformations be linear in all the variables
(x,y,z,t,x',y',z',t'). That
was handled by the statement that we would _first_ seek a set of
transformation equations which was linear, but if
necessary we could back off
from that requirement.

Briefly, if you let

x = A x' + B y' + C z' + D t', y = E x' + ..., etc., the
unknown constants can
be identified by symmetry arguments and the requirement
that the speed of
light be measured by both primed and unprimed observers
as c. The Lorentz
transformation falls out in your lap, i.e., the Lorentz
equations can indeed
"be deduced from A" + the additional requirement that the
dependence be
linear.

In his "Relativity: The special and general theory"
Einstein does not postulate
linearity but almost explicitly introduces a third axiom:

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).

Physically, the third axiom sounds silly but mathematically
it is equipollent to
the second - it is indispensable for the determination of
initially unknown
parameters.

Wild. But couldn't 3 be rephrased as:

3. If and only if in a reference frame S the speed v of a
reference frame S'
(x = vt) is equal to the speed of light, then the speed of
light is zero in S'
(x' = 0). But if v = c, then gamma = infinity, we'd have the
product of
infinity and zero in Lorentz transformation. Basically all
lengths are
contracted to zero, all positions and times measured as zero
in the frame
moving at the speed of light. Since this only deals with a
limiting case, I
really can't see this as a viable postulate on the same level
as the two
canonical postulates of special relativity.

Clearly, axioms 2 and 3 are, independently, corollaries of Lorentz
equations.

Yes, 2 can be derived from the Lorentz transformation, and
it's great fun to
show that the wave equation for E-M waves is invariant under a Lorentz
transformation -- the laws of physics are the same for all
observers, etc.
But the Lorentz transformation is not sufficiently intuitive
for me to be
happy with treating it as an axiom.

For the moment, it seems to me that Lorentz equations CAN be derived
from the three axioms, but I am still not sure - there may
be some hidden fourth
axiom. In any event, Lorentz equations cannot be derived
from axioms 1 and 2
alone.

Has that actually been shown? Granted that I've only seen it
done with the
additional linearity condition (or first-order linearity, as
John S. Denker
has pointed out), has some _non-linear_ tranformation been
found that obeys
axioms 1 & 2? That would show that the Lorentz
transformation is not uniquely
determined by the two axioms of special relativity. Or maybe
an argument can
be made that first-order linearity is "clearly necessary"
(for what reasons?)
and so can be assumed to be fundamental?

It may just be a case for Occam's Razor. It'd take linearity
as an assumption
long before #3 above!

Ken