Re: [Phys-L] How Einstein discovered E = mc2

[John Denker <jsd@av8n.com>]

On 11/20/2014 12:47 AM, Savinainen Antti wrote:
> I read two papers by Eugene Hecht [....]
>
> <http://scitation.aip.org/content/aapt/journal/tpt/50/2/10.1119/1.3677283>
> <http://scitation.aip.org/content/aapt/journal/ajp/79/6/10.1119/1.3549223>

FWIW, this is the same Hecht who wrote several textbooks (and
outlines) on physics and optics.

I am mystified by the two AJP articles cited above.  The E=mc^2
formula reminds me of certain tabloid celebrities who are
"famous for being famous" even though they've never accomplished
anything of significance.

Anybody who is serious about physics moved on to other things,
long since.

Specifically:
  a) Suppose we have a pointlike particle, and we think 
   its position is a 4-vector (r).  This is not the only 
   conceivable way of doing things, but it seems to work.

  b) Then for any particle with nonzero mass you can define 
   the 4-velocity u = dr / dτ.  This is far from the only way 
   of doing things, but it is as good as any and better than
   most.  It is nice and simple.  It upholds the correspondence 
   principle.

  c) Then you can define the 4-momentum p = m u (for some "m").
   This is exceedingly simple.  It upholds the correspondence 
   principle.  As far as anybody can tell, it is consistent 
   with conservation of momentum.

  d) For massless particles you can skip steps (b) and (c)
   and just write down the 4-momentum.  These particles have
   undefined 4-velocity and undefined proper time, but they
   have a perfectly good 4-momentum

  e) In any case, m is the invariant length of the 4-momentum.
   For massless particles, m exists, and is equal to zero.

  f) Suppose we pick out the timelike component of the
   4-momentum in some chosen frame, and choose to call it
   the "energy".  Then by the basic definition of norm of
   a 4-vector, we have
        E^2 - p_xyz^2 = m^2              [1]
   where E is the timelike part of the momentum and p_xyz
   is the spatial part ... both of which are definable only
   with respect to some chosen frame.  This stands in contrast
   to m, which is invariant and frame-independent.

  g) For something that is not a pointlike particle, perhaps
   a parcel of fluid, all of the above goes out the window
   and we need heavier tools, such as stress-energy tensors.

My point is that equation [1] is vastly more useful than
E=mc^2 ... and also conceptually simpler, since it is a
routine consequence of other things we know.  E=mc^2 is
not the miracle, or even a miracle.  Spacetime geometry 
and trigonometry is where the action is.

Hecht argues that nobody has ever come up with a rigorous
proof of the "equivalence" of mass and energy.  Well of 
course not, because they are not equivalent.  They might 
be numerically equal in certain limiting cases, but they 
are not "equivalent".

There will never be a rigorous answer to the question,
because there is not even a rigorous question to the
question.  Einstein himself remarked:  "As far as the 
laws of mathematics refer to reality, they are not certain, 
as far as they are certain, they do not refer to reality."

The classical energy is not defined uniquely;  you can 
shift it by a gauge transformation.  Therefore you can
always pick a gauge so as to /make/ the classical energy 
equal to mc^2 -- which an awful lot of people have done -- 
and it would even be consistent with the correspondence
principle.  However, you could equally well make it equal
to (mc^2 / 2) or any other thing that suits your fancy.
You need to know more than the correspondence principle
in order to do physics.

Seriously, you could define "the" velocity to be
 -- v = dr / dt         (t = coordinate time)
 -- u = dr / dτ         (τ = proper time)

and both would reduce to the classical velocity in the 
correspondence limit.  Now v has the advantage of being
easy to measure, and of existing even for massless
particles (whereas u does not) --  but the other 99% of
the time, u is more useful.  So there will never be a
unique answer to the question of how to generalize our
classical notion of velocity.

By the same token, there will never be a unique answer
to the question of how to generalize our classical notion
of mass.  Classically mass serves as a measure of "inertia"
(whatever that means) and also serves as a source term
for the gravitational field.  When we generalize to
relativity, you can't have it both ways.  Conventionally
we take m to be the "inertia" constant in p = m u, in
which case m is no longer the source term for the
gravitational field.  The stress-energy tensor takes 
over as the source term.  You have to be super-careful
how you state the equivalence principle (the equivalence
of gravitational mass and inertial mass), because any
bold general statement would be untrue.

Even without mentioning gravitation, just confining
ourselves to "inertia", if you try to define a
velocity-dependent relativistic mass, you quickly
discover that there are lots of inequivalent ways
of doing it:  longitudinal mass, transverse mass,
and who-knows-what else.

So you see, we can't even say that mass is equivalent
to mass in all the ways we might like, so there's just
no hope of energy being equivalent to mass as a general 
proposition.

There are relationships among these things, but the
relationship is not an "equivalence".

Saying such things are equal in the correspondence limit
is nice, but it's sort of like checking the dimensions:
You have to check, but lots of things have the correct
dimensions even when they're not the right answer.  The
torque has the same dimensions as the lagrangian, but
they're not the same thing.


Last but not least:

> honest historical research instead of portraying a smoothed version
> of the process of discovery

There are at least two common mistakes.  As mentioned above,
one mistake is to smooth out the history, pruning off all 
the dead ends, pretending that discovery is not nearly so 
messy and non-monotonic as it really is.

Another is to /personalize/ the history.  I say the 1860s 
were not all about Abraham Lincoln, and ancient Greece 
was not all about Pericles.  By the same token, relativity
is not all about Einstein.  Really, really not.  I do not
much care what Einstein was up to in 1908;  I am much more
interested in what Minkowski was up to.

Some people say that personalizing the story makes it more
interesting to students.  Well maybe so, but it comes at a
terrible price.  Nobody wants to get involved with a project
if they think the famous guy is going to get all the credit.
This is bad for everybody, including the famous guy along
with everybody else, because it makes it hard to form teams.
This business of giving Einstein credit for stuff that other
people did has got to stop.