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Re: special relativity



Regarding Justin Parke's question:

Most textbooks and teachers (at least all that I have seen or known)
introduce special relativity by introducing two postulates:

1) The laws of physics are the same in all reference frames

2) The speed of light is independent of the speed of the source and
the observer.

I have always thought that these are redundant. Isn't 2) implied in
1)?

First, these 2 postulates do not exhaust the set of postulates needed
for special relativity. These postulates do not in themselves
specify the form the relativistic laws of mechanics (relationship
between velocity and relativistic mechanical momentum, relationship
between momentum and kinetic + rest energy, and the like. What these
postulates *do* imply is that Lorentz transformations are the
physically proper ones to use when transforming space-time
displacements from one inertial frame to another one. Once we have
Lorentz transformations they imply the usual relativistic effects
such as the formulae for time dilation, length contraction, and for
the addition law for the composition of velocities.

Second, although Einstein's 2nd postulate (combined with the first
one) *do* imply Lorentz transformations, it is *not* the most
logically compact and elegant formulation of the theory. There
really is the necessity for 2 postulates (since the first
postulate 1 by itself is also compatible with Newtonian physics if
Galilean transformations are used to transform between inertial
frames in relative motion in mechanics. The most logically
satisfying version of the 2nd postulate is that:

2) There are no instantaneous interactions at a distance in
physics, (and consequently, there exists an upper bound, i.e. a
speed limit, on the rate at which causally informative influences
can propagate from an event at one spacetime point to another one
physically separated from it in space).

The fact that this speed limit of causation has the same value in
all inertial frames is a *theorem* from the first 2 postulates.

The forms of the laws of relativistic dynamics (i.e. energy,
momentum, & velocity interrelationships) for particles are dictated
by the first 2 postulates above *plus* a *third* postulate, which
effectively states that the laws of classical mechanics satisfy
Hamilton's Principle of least (er, stationary) action. Once we
have Hamilton's Principle and have the other 2 postulates the
usual formulae for special relativity follow (for motions observed
in inertial frames). A couple of important consequences of these
laws are that all particles with a positive (rest) mass *must*
travel at speeds strictly *slower* than the speed limit c--no matter
how high their momentum and kinetic energy, and also particles that
have a strictly *zero* value for their (rest) mass *must* travel *at*
the speed limit c no matter how low their momentum and energy. The
direction of the velocity vector and the (mechanical) momentum
vectors are *always* parallel to each other regardless of whether or
not the particle's mass is zero or not. The fact that neither
massless *nor* massive particles travel *faster* than c is a direct
consequence of (the corrected) postulate 2.

The fact that light happens to travel *at* this speed limit of
causation is a theorem that depends on the observed masslessness
of the photon and on relativistic dynamics. Since it is a theorem it
is not a good idea to claim it is a postulate. Besides
electromagnetic waves, other massless things that travel *at* the
speed limit c are gluons, and gravitational waves. Until recently
it was within the realm of experimental (and theoretical) possiblity
that neutrinos, too, traveled *at* the speed limit. Since any
massless object travels at this speed limit it is probably not a
good idea to call this speed limit 'the speed of light' as if light
was the only conceivable thing that could go at that speed.

If photons happened to have a positive (rest) mass then it would
mean that (the group velocity of) EM waves in a vacuum would travel
*slower* than c with a speed that was dependent on the frequency.
The validity of special relativity is independent of the issue of
whether or not light always travels at speed c in all inertial
frames. That is another reason for not taking the constancy of the
local speed of light as a postulate of relativity. Such a postulate
would needlessly mess up relativity if it was ever discovered that
photons, indeed, do have a tiny mass. If such a discovery is never
made then the main argument against using Einstein's 2nd postulate
as it stands is an esthetic and logical one related to the
logical structure of the theory.

David Bowman