The Path To Mass-Energy Equivalence

Edward Green wrote...
 I'll wager nobody here can show me in a few sentences the 
 logical path extending from the simple principles of relativity 
 to the famous mass-energy equivalence relation...

The equivalence of mass and energy is not a logically necessary 
consequence of the simple principles of special relativity.  It 
is, however, strongly suggested by those principles - one of the 
first (of many) examples of the remarkable heuristic power of 
Einstein's theory.

We postulate that the natural measures of spatial and temporal
intervals in any and every inertial frame are such that the
velocity of light with respect to those measures is invariant.
This immediately implies that relative velocities are not
transitively additive from one reference frame to another, 
and, as a result, the acceleration of an object with respect 
to one inertial frame must differ from its acceleration with 
respect to another inertial frame.  However, by symmetry, the 
FORCE exerted by two objects upon each another is equal and
opposite, regardless of their relative velocity.

So, given an object O of mass m, initially at rest, we apply 
a force F to the object, giving it an acceleration of F/m.
After awhile the object has achieved some velocity v, and we 
continue to apply the constant force F.  But now imagine 
another inertial observer, this one momentarily co-moving 
with the object at this instant with a velocity v.  That other
observer sees a stationary object O of mass m subject to a 
force F, so, on the assumption that the laws of physics are 
the same in all inertial frames, we know that he will see the 
object respond with an acceleration of F/m (just as we did).

However, due to non-additivity of velocities, the acceleration 
with respect to OUR measures of time and space must now be 
different.  Thus, even though we're still applying a force F 
to the object, its acceleration (relative to our frame) is 
no longer equal to F/m.  In fact, it must be less, and this 
acceleration must go to zero as v approaches the speed of 
light.  Thus the effective inertia of the object increases 
along with its velocity.

During this experiment we can also integrate the force we 
exerted over the distance travelled by the object, and determine 
the amount of work (energy) that we imparted to the object in 
bringing it to the velocity v.  With a little algebra we can 
show that the ratio of the amount of energy we put into the 
object to the amount by which the object's inertia (units of 
mass) increased is exactly c^2.

From these considerations Einstein formed the hypothesis
that ALL inertia is potentially convertable to energy, but
clearly this doesn't follow rigorously from special relativity.
It was just a hypothesis *suggested by* special relativity
(and also Maxwell's equations).  At the time (1905) the only 
experimental test Einstein could suggest was to see if a lump 
of "radium salt" loses weight as it gives off radiation, but 
of course that would never be a complete test, because the 
radium doesn't decay down to nothing.  The same is true with 
an atomic bomb, i.e., it's really only the binding energy of 
the atoms (or nucleus for a hydrogen bomb) that is being 
converted, so it doesn't demonstrate an entire electron or
proton (for example) being converted into energy.  However, 
today we can observe electrons and positrons annihilating 
each other completely, and yielding energy in precisely the
amounts predicted by Einstein in 1905.