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*From*: John Denker <jsd@av8n.com>*Date*: Thu, 20 Nov 2014 17:50:11 -0700

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.

**Follow-Ups**:**Re: [Phys-L] How Einstein discovered E = mc2***From:*Moses Fayngold <moshfarlan@yahoo.com>

**References**:**[Phys-L] How Einstein discovered E = mc2***From:*Savinainen Antti <antti.savinainen@kuopio.fi>

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