You have made a mistake here- by guessing that your physics is probably wrong. The rest of the argument is fine.
Some people quibble about the use of the word 'mass' but the
essential physical fact is that a box of photons will indeed cause as
much gravity as a box of coal with the same total energy. I use a box
so that the net momentum of all the photons will cancel and we can just
consider their energy.
Mike W.
The version practitioners use instead of E=mc^2 is this one:
E^2 = (mc^2)^2 + (pc)^2
where "m" is what some people call "rest mass" and others the
"invariant mass", and is zero for a photon. The variable p is the
momentum. Nonrelativistically, p=mv, but this doesn't work
relativistically, and the problem is very noticeable in the case of a
photon. The formula above describes the relationship between energy,
momentum, and mass, and is valid in all frames of reference.
Unfortunately there is no frame which moves with a photon
(there is no frame in which a photon is at rest).
Mike's box of photons is a great example because in general there
will be a frame in which the sum of all the momenta of the photons in
the box is zero. It turns out the above relationship between E, p, and
m works also for combinations of particles. Add up the energies, add up
the momenta (as vectors -- they have x, y, and z components), and you
can compute the invariant mass of the collection of particles. It's
nice this works out because most objects we work with are made up of
component parts. It's nice that we can think of an atom or a block
moving relativistically without having to worry about each subatomic
particle inside. We don't even know if the particles we think of as
fundamental aren't made up of pieces themselves.
Tom
(published on 10/22/2007)
