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

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding the part of Ken Caviness' post where he wrote:

> 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

I have stayed out of these foundations of relativity discussions
until now for fear that my patience would have been overtaxed had I
jumped in.  However, I think I can safely make a few comments
motivated by Ken's intriguing comment above.

Ken is correct that Einstein's 2nd postulate of the constancy of the
speed of light is a corollary, but it depends on more than just the
1st postulate of the physical equivalence of all inertial reference
frames.  It depends on the actual 2nd postulate of relativity as
well.  The first postulate is consistent with *both* special
relativistic *and* Newtonian physics.  What distinguishes them is
the true 2nd postulate.  This 2nd postulate has multiple forms
(not all that different from the many forms of the 2nd law of
therm, but not nearly *that* many different forms).  The
version of the 2nd postulate of special relativity is that there
is no instantaneous interaction at a distance.  This is in sharp
constrast to the analogous 2nd postulate of Newtonian physics
that all observers always observe common time intervals between
pairs of events.

If one takes the 1st postulate of the equivalence of all inertial
frames *and* takes the 2nd assumption of a common time interval
for any pair of events for all observers we can deduce that all of
those equivalent inertial frames are related to each other by
Galilean transformations.  OTOH, if we replace this 2nd postulate
with the SR postulate of there not being any instantaneous
interactions at a distance we instead deduce a number of important
results, such as that a speed limit of causation exists (let's call
it c), that that this speed limit c is the same for all inertial
coordinate systems in all directions and at all places, and that all
the different inertial frames are related by Lorentz transformations.
Actually, once we include the effects of the possible affine shifts
in spatial and temporal origin all the different inertial coordinate
systems are related by Poincaré transformations which are
parameterized by a noncompact 10-parameter continuous group of
transformations.  Only the homogeneous subgroup of the Poincaré
group are the Lorentz transformations.  And of these only those which
are continuously connected to the identity transformation comprise
the 'proper' Lorentz group, where the others involve things like
time reversal and or spatial inversions and reflections, etc.  Of the
Proper Lorentz transformations the maximal compact subgroup
consists of the set of all proper rotations.

It is interesting (and an important source of real physics, i.e.
Thomas precession) that the noncompact equivalence class of all pure
boosts do not form a subgroup of the proper Lorentz group.  This is
because a composition of two different pure boosts in two different
directions does not result in another pure boost.  The result is
typically actually a combination of a boost *and a rotation*.

These two postulates determine the form of the Lorentz
transformations and all their usual effects like time dilation,
length contraction, and the non-simultaneity of spacelike pairs of
events across different inertial frames, expressions for proper
time interval.  Also such things as the invariant trichotomy of all
pairs of events being either time like, spacelike, or null, etc.
along with there being a scalar invariant interval measure between
them are also results of these two postulates.  To get the result that
the speed of light is a constant in all inertial frames and to get
that this speed is the same speed c as the speed limit of causation we
need to supplement these two postulates with a 3rd postulate which
fixes the form of the dynamical equations of motion and the
relationship between such things as mass, momentum, velocity, energy
etc. in SR.  The simplest form of this 3rd postulate is that
dynamics obeys Hamilton's Principle of least action.  Once we have
this postulate the results of the usual relativistic formulae
connecting m, p, E, v, L, etc. hold, and such quantities obey the
usual conservation laws for E, p, & L.  Also, we can see that in the
limit of c --> [infinity] these formulae boil down the the usual
Newtonian expressions.

A couple of the important consequences of Hamilton's principle for
free particles combined with the implications of the other two
postulates (namely that different inertial coordinate systems are
related by Poincaré transformations) are that 1) particles with a
positive mass must always travel slower than speed c, with their
lowest energy being a state of rest, (i.e. no momentum), and 2)
that particles with exactly zero mass must travel *at* speed c
independent of their energy and momentum (which happen to
have the energy and the magnitude of the momentum being directly
proportional to each other with c being the proportionality factor).
Once we experimentally determine that photons of light happen to
be massless we find that they must travel at speed c (which we
already found means that that speed is a constant in all inertial
frames in all directions at all places).

If we also experimentally find that photons have a spin 1, and
are uncharged along with them being massless we find that their
classical wave properties must be described in terms of
Maxwell's equations which are themselves invariant under
Lorentz transformations.

Note that the form of the 2nd postulate of relativity that I gave
above must either forbid the existence of tachyons, or if it does
allow them, it must require that tachyons can have no causal or
detectable influences on anything that we can measure (i.e.
non-tachyonic matter).  Now it *is* possible to somewhat relax the
2nd postulate slightly to a form that would allow the existence of
tachyons and allow them to interact with non-tachyonic matter.  But
such a relaxation of this postulate would have a problem that is
aesthetically unappealing to me.  The modified form of the 2nd
postulate that would allow interacting tachyons is to not require
that there actually be no instantaneous interaction at a distance,
but to, instead, require that there exist a physical speed (namely c)
that has the (invariance) property that any motion with that speed in
any inertial frame will also have that exact same speed in any other
inertial frame as well.  Only motions with this speed have this
invariance property.  All other motional speeds can result in
different values for the speed of the motion in different inertial
frames.  Using this alternative form of the 2nd postulate gives all
of the previous results of SR, but it also allows tachyons as well &
and allows them to be detected by our instruments as well.  The
aesthetic down side to this alternate form (to my mind) is that there
is no prior physical motivation for wanting there to be such an
invaraint speed c in nature.  This is just pulled out of a hat.  At
least the 'no instantaneous interaction at a distance' postulate has
a nice motivation and interpretation in terms of retarded causality,
which makes lots of sense to me.

David Bowman