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Re: c



Jack U. wrote:
But how would we know that "c" represents a limiting speed?
By hypothesis. The existence of a speed limit is intimately tied to the
requirement that there can be no instantaneous-action-at-a-distance (IAAAD).
If such a limiting speed didn't exist then there would be no bound on how fast
a cause "here" can influence an effect "there", and this would get us back to
an arbitrarily fast propagation of causal influences which would then
contradict the requirement of no IAAAD. The label "c" just is the name given
to this limiting speed.

To
say that a photon mass "is not sufficient to invalidate special relativity"
does not respond to the deeper question, "Is special relativity a
consequence of the facts that (1)there is a limiting speed at which signals
can be transmitted [I carefully stipulated that the photon mass is the
smallest of all - no massless neutrinos] and (2) signals actually can be
transmitted at that speed?"
The answer is "yes" for #1, and "it is not absolutely necessary, but it sure
helps, and makes things a lot easier" for #2.

The Landau/Lifshitz derivation starts (Sec 1-2) with a definition
of "intervals". They then go on to say (p.4 of the 1951 edition) "we
now express the invariance of the velocity of light in mathematical form",
having introduced (in modern language) ct as a "4th dimension". They
then proceed to obtain the Lorentz transformation in a very conventional
way by requiring the the "interval" between 2 events is an invariant.
With a lawyer's skill your quote deftly skips over the preceding discussion
which starts with the first word at the beginning of the book on page 1. This
out-of-context quote gives the impression that L&L do things in the usual
manner regarding the constancy of the velocity of *light* as what is
fundamental here, rather than the idea that there is no IAAAD and that a
causal speed limit must exist. On p.1 L&L discuss the untenability of IAAAD
and then say:

"However, experiment shows that instantaneous interactions do not exist in
nature. Thus a mechanics based on the assumption of instantaneous propagation
of interactions contains within itself a certain inaccuracy. In actuality, if
any change takes place in one of the interacting bodies, it will influence the
other bodies only after the lapse of a certain interval of time. It is only
after this time interval that processes caused by the initial change begin to
take place in the second body. Dividing the distance between the two bodies
by this time interval, we obtain the *velocity of propagation of the
interaction*.

"We note that this velocity should, strictly speaking, be called the
*maximum* velocity of interaction. It determines only that interval of time
after which a change occuring in one body *begins* to manifest itself in
another."

L&L continue with this discussion for a while. Later on p. 2 they say:

"From the principle of relativity [this means the 1st postulate, D.B.] it
follows in particular that the velocity of propagation of interactions is the
*same* in *all* inertial systems of reference. Thus the velocity of
propagation of interactions is a universal constant. This constant velocity
(AS WE SHALL SHOW LATER,[emphasis, D.B.]) is also the velocity of light in
empty space. The velocity of light is usually designated by the letter c, and
its numerical value is c = 2.998 x 10^10 cm/sec."

Notice that L&L show *later* (after they demonstrate that massless particles
must move at the speed limit and massive ones must move more slowly) and that
light travels at the speed limit. Since L&L shortly acquiess to common usage
they call this speed limit "the velocity of light" later on in their
discussion and also in the quote above that Jack gives. In their discussion
in section 2 on intervals, L&L show that the two postulates of: 1. invariance
of physical law in different inertial frames, and 2. the universal causal
speed limit c, that an invariant interval *must* exist between different
"events" which has the same value in all inertial frames. This derivation
does not use any of the special properties of light other than that it
conveniently happens to propagate at the universal speed limit. The
discussion could have just as well used signals with neutrinos as light. In
principle, the derivation would work with massive particles carrying the
signals, but then each event sequence would have to be repeated many times
with evermore energetic signals and a limiting process taken. Needless to
say, this is *quite* cumbersome in practice and things work much easier if
one is allowed to use signals which move at the speed limit. After L&L derive
the invariant interval, their subsequent disscussion does follow the
conventional route of deriving the Lorentz transformation from the requirement
that the interval must really be invariant among different inertial frames.
After LTs are derived, the relativistic dynamics of particles is derived
using a Lagrangian formulation of Hamilton's principle with the constraint
that the eqns. of motion derived must satisfy the 1st postulate of relativity,
and that the different frames are related to each other by LTs.

So it seems to me that your answer is on a different logical
plane than the question that I posed. I thought that my question invited
some speculation on the interconnectedness of everything, and that a
universe with no massless particles would be much different from one
with just a few patches placed on E&M, as you have described.
I'm sure that in a universe where all the particles had a mass whose rest
energy was greater than 1 keV would be a much different place than the one we
find ourselves in. OTOH, one which has all particle masses greater than
10^-50 kg, or equivalently rest energies greater than 6 x 10^-15 eV would be
indistinguishable from the one we currently have (assuming that all the
ordinarily massive particles keep their usual masses).

Your appeal to gravity waves as the carrier of signals at the
ultimate velocity is intriguing. Maybe someone can dream up a question
that probes the necessity for special relativity in this context.
You may have me here. (The suggestion of using gravity waves was with the
implicit proviso that the nonlinearites of GR would not mess up the SR
theory and that the gravitons would be thought of as just another massless
field.) The gravity waves would have to propagate on a space-time between
massive sources which is well-approximated as flat Minkowski space and whose
amplitudes are weak enough so the self-interaction from the nonlinearity of
Einstein's equations that the gravity waves can effectively propagate as free
massless fields. The problem is to have sources massive enough to create and
respond to the waves in the first place, which themselves do not violate the
weak field condition on the spacetime around them. This, admittedly, would be
a neat trick considering the intrinsic weakness of the gravitational
interaction. One possible out would be to use gravity waves for signaling in
special relativity *only* in universes whose law of gravity is something like
the Nordstrom's scalar theory. In this case things can be set up so that
special relativity remains valid in the presence of gravitational fields. I
think that it would be cheating to appeal to the local validity of SR in the
GR context, however, to set up signaling situations to use in finding the
proper form of the local LTs when using gravity waves for the signals. I
prefer in this instance (of gravity wave signalling) to work in a universe
where gravity is just another interaction which doesn't mess up geometry. :)

David Bowman
gtc.georgetown.ky.us

P.S. Since I am getting weary of this subject I plan to lay off it from now on
for a while. Extended discussions of topics are hard on us verbose writers.
I'm sure it isn't easy on our readers either. :)