Tangled Up In Heuristic

Edward Green wrote
 Let me try to summarize the argument:
 (1) We postulate (or define) the continued validity of 
     Newton's third law.
 (2) We therefore have a frame invariant meaning for force (?).
 (3) We therefore have an unambiguous measure for the work done 
     on an object in accelerating it.
 (4) If we want to preserve Newton's second law also, we are 
     led to the concept of relativistic mass.
 (5) The ratio of work done on an object to the increase in
     relativistic mass is found to be c^2.
 (6) We postulate that mass/energy equivalence in this ratio 
     is a general feature of the world, not just germane to 
     kinetic energy.

I'd concur with this as an accurate summary of the historical
development, although for a modern treatment we could tinker a bit
with the sense and order in which Newton's 2nd and 3rd laws are
invoked.  Naturally there are many different (though physically
equivalent) ways of formulating mechanics in special relativity, but
the most common these days is to begin by defining momentum as the
product of (rest)mass and velocity (in accord with Newton's
Principia).  The motivation for this definition is that conservation
of this 3-vector is well-behaved under Lorentz transformations (i.e.,
if it is conserved with respect to one inertial frame then it is
conserved with respect to all inertial frames), and it agrees with
non-relativistic momentum in the limit of low velocities.  Then we
essentially postulate that this "momentum" is, in fact, conserved
(extrapolating from our experience with momentum at low velocities).

(It's interesting that we use this "correspondence principle" in
defining our observable parameters in relativity, just as we do in
quantum mechanics, but apparently in the historical development of
relativity no one dignified this heuristic with a name, let alone
elevated it to a "principle"; the latter move was characteristic of
Bohr.)

Then, once again following Newton's Principia, we define
(relativistic)force as the rate of change of momentum.  This is
Newton's 2nd law, and it's motivated largely by the fact that this
"force", together with conservation of momentum, implies Newton's 3rd
law, at least in the case of impacts (and we can build up many other
kinds of interactions by imagining the exchange of various impacting
intermediate bodies).

You might notice that although we've adhered closely to the
Principia's original definitions, we've actually reversed the order
of development, because Newton postulated the 3rd law and then
derived conservation of momentum, whereas we did it the other way
around.  In my earlier post (as you rightly summarized) I reverted
more to Newton's arrangement, taking the 3rd law as more immediate. 
Speaking of which

Edward Green wrote
 ... it is not obvious to me that 'by symmetry' we must still
 require the force between two bodies in relative motion to 
 be equal.  ...If we took the force on the moving body to be 
 different from the body at rest then 'symmetry'... would 
 require that the magnitudes of the two forces be exactly 
 reversed if we shifted to the other body's frame; but this 
 does not by itself require equality.

You've invoked a distinction here between "the moving body" and "the
body at rest", but according to the principle of relativity there 
is an equivalence between any two inertial frames, so they're to be
regarded as symmetrical.  From this and the principle of sufficient
cause, the 3rd law is pretty much unavoidable.  Also, it supports our
conservation of momentum, as mentioned above.  Admittedly there are
some complications when applying the 3rd law to extended interactions
in a relativistic context, but for coincident points of mutual
contact it would result in a very convoluted theory to define force
in an asymmetric way, without conflicting with the physical
equivalence of inertial frames.

Of course there's another assumption lurking here as well, namely,
the assumption of physical equivalence between instantaneously
co-moving frames, regardless of acceleration.  For example, we assume
that two co-moving clocks will keep time at the same instantaneous
rate, even if one is accelerating and the other is not.  This is 
just a hypothesis - we have no a_priori reason to rule out physical
effects of the 2nd, 3rd, 4th,... time derivatives.  It just so
happens that when we construct a theory on this basis, it works
pretty well.  (Similarly we have no a_priori reason to think the
field equations necessarily depend only on the metric and its 1st 
and 2nd derivatives; but it works.)


Edward Green wrote
 In special relativity 'a' is _not_ frame invariant, so that 'f' 
 and 'm'cannot both be frame invariant anymore either, but one at 
 least must vary.  Evidently we choose 'm' (pace some comments that 
 'relativistic mass' is no longer a fashionable concept?).

The subject of "relativistic mass" is kind of tricky.  On one hand,
you're right that most modern formulations of relativity shun this
concept, but on the other hand it's undeniable that the concept was
heuristically significant in suggesting the equivalence of mass and
energy.  On the third hand, it's worth noting that Einstein's 1905
paper was entitled "Does the INERTIA of a Body Depend on its Energy
Content?", so clearly Einstein was mindful of some distinction
between inertia and mass.  On the fourth hand, not only is
mass-energy equivalence not required by special relativity, it is
actually inconsistent with it when combined with the equivalence
principle, and this is one of the main reasons that Einstein
eventually abandoned SR.  Thus, SR led Einstein to mass-energy
equivalence, which in turn led him to reject SR!  A similar thing
could be said of "relativistic mass".

On the fifth hand, it's certainly more elegant to introduce the
mass-energy relation by means of the energy-momentum 4-vector, but
some people find that the mathematical elegance obscures the physical
content.  To illustrate, take m as the constant rest mass, and define
relativistic force as the four-vector

                       d^2 x_j
               f_j = m -------
                       d tau^2

where j = 0,1,2,3, and x_0 is the time coordinate.  Then define the
energy-momentum 4-vector as

                          d x_j
               p_j  =  m  -----
                          d tau

from which it follows that

                        d p_j
                f_j  =  -----
                        d tau

As mentioned above, we use our correspondence principle to decide
that we should define momentum as the 3 space components of p_j. 
This just leaves the time-component p_0, which we choose to define as
the energy E.  Thus,

                              d x_0
             E  =  p_0  =  m -------
                              d tau

Recall that x_0 is just our coordinate time parameter t, which is
related to the object's proper time tau according to
                          _____________
               d tau  =  / dt^2 - dx^2

                             ______________
                      =  dt / 1 - (dx/dt)^2

Note that we're using geometric units here, so dx/dt = v/c can just
be called v.  Substituting back into our energy definition (and
recalling that x_0 is simply t) we have

                  m
      E  =  -------------  =  m + (1/2)mv^2 + (3/8)mv^4 + ...
            sqrt(1 - v^2)

The first term, "m" is now interpreted as the rest energy of the
mass, but did we actually "deduce" this?  Notice that we violated our
sacred "correspondence principle" in the definition of E, because by
correspondence with the low-velocity limit the energy of a particle
SHOULD be something like (1/2)mv^2, but clearly the time component of
p_j does not reduce to that in the low-speed limit.  Nevertheless, we
defined p_0 as the "energy" E, and since that component equals m 
when v=0, we essentially just DEFINED our result E=m (or E=mc^2 in
ordinary units) for a mass at rest.  It isn't clear that this is
anything more than a bookkeeping convention, one that could just as
well be applied in classical mechanics using some arbitrary squared
velocity to convert from units of mass to units of energy.  The
assertion of equivalence between mass and energy has physical
significance only if it is actually possible to effect a conversion
from one to the other, and the only suggestion given to us by 
special relativity for such actual conversion is the increase of 
a "mass-like" quality (i.e., inertia) as we do work on (i.e., add
energy to) an object.

Of course, the fact remains that this mass-like quality (aka
relativistic mass) is distinct from rest mass, but we'll probably
need a more fundamental theory of matter to understand the actual
convertability between the two.  No one really knows the detailed
process by which rest mass is converted to energy.  Take for example
an atom of some highly fissionable material.  We may split that 
atom into two smaller atoms whose combined rest mass is less than 
the original rest mass, but at the instant of the split the overall 
"mass-like" quality is conserved, because those two smaller atoms 
have enormous velocities, precisely such that the total relativistic
mass (archaic or not) is conserved.  Then we slow down those two 
smaller atoms and end up with two atoms at rest, at which point a 
little bit of rest mass has disappeared from the universe.  But 
the actual physics of how the excess binding energy was originally 
a "rest property" representing "real mass" with isotropic inertia, 
and then becomes a kinetic property representing archaic old 
relativistic mass with anisotropic inertia, is not well understood 
(at least not by me).