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Re: [Phys-l] E=mc^2 because E=mc^2?



On 04/23/2007 10:13 PM, Michael Porter wrote:
Scientific American's on-line "Ask the Expert" has a Kwik'n'Eezy derivation of E=mc^2:

<http://www.sciam.com/askexpert_question.cfm? chanID=sa005&articleID=1F16687F-E7F2-99DF-39477FE5DA7426E2>

The first equation they use in the derivation is p(photon) = E/c. Doesn't this assume E=mc^2? It looks like circular logic to me.

That part of the argument is circular, and other parts are
even worse.

1a) A circular argument doesn't prove anything, and passing it
off as a proof or a derivation is unacceptable.

1b) OTOH, a circular discussion of facts that are /already/
known to be true is sometimes useful as a consistency check
or as a mnemonic.

2) The sciam article is even worse than that, because it mixes
some untrue ideas into the argument, such as the assertion:
>>> Therefore the photon must have a mass, m.

At this point we get into some interesting physics:

A) The correspondence principle tells us that special relativity,
when applied to a nonrelativistic situation, must make the same
predictions as classical (pre-relativistic) physics.

The converse does not hold! You cannot take classical ideas and
apply them willy-nilly to relativistic situations.

A perfect example is coordinate time (t) versus proper time (tau).
In the nonrelativistic limit, they reduce to the same thing, and
classical physics does not make the distinction. Relativity
most certainly does make the distinction.

B) The law of conservation of momentum is important. The special
relativistic law tells you (via the correspondence principle)
the form of the nonrelativistic law (and not vice versa).

C) In a nonrelativistic situation, we can make various statements
about the behavior of the center-of-mass of a system. These can
be seen as immediate corollaries of the conservation of momentum
/and/ the assumption that mass is unchanging.

These statements do not automatically carry over to special
relativity!

In particular, if we define "mass" the way thoughtful experts
do (and have done for many decades), then photons are massless
and it is trivial to come up with examples where the center of
mass hops around, even in the absence of externally-applied
forces. Just convert massive electron-positron pairs to
massless photons.

When somebody starts making center-of-mass arguments involving
the mass of photon, it's hard to take the argument seriously.


Constructive suggestion:

You are much better off relying on /conservation of momentum/
than on center-of-mass arguments.