Re: A small conservation of energy puzzle

On Thu, 02 Feb 2006 10:54:52 -0800, PD wrote:


sal wrote:
Here's a little problem in conservation of energy which I found rather
puzzling. For your amusement...

(C=1 throughout, and so E=M throughout. It's a simple problem but a
precise statement takes about a page; bear with me...)

We have two frames of reference, a spaceship's frame, "M", and an
Observer's frame, called "O".

On a spaceship, an astronaut converts a small mass, "dM", into a single
photon of frequency "f", which zips away from the ship at C.

A single chunk of mass cannot convert into a single photon without
violating conservation of momenergy. This is why that doesn't happen.

You're thinking of the case where there is an isolated mass which is
completely converted into photons. That's not what we have here.

In this case there's a spaceship present, with mass >> dM (spaceships
generally mass a lot more than a single photon, though I never stated
that explicitly). A large mass can certainly emit one photon without
violating conservation of momentum.

With total conversion, as in matter/antimatter annihilation, the photons
come out in matched pairs going in opposite directions (in the emitter's
rest frame). But for a mundane example where that "pairwise cancellation"
of momentum obviously doesn't happen, consider a flashlight. When you turn
it on, the batteries start losing mass as they run down, and that mass is
converted to photons. But the photons all come squirting out one end,
with nothing coming out the other end to "balance" them. The problem
statement could equally well have said, "An astronaut turns on a
flashlight. The batteries are almost dead and just one photon comes out,
which goes zipping away from the ship..."

The reason momentum balances in this case is, of course, the recoil of the
large mass (flashlight or spaceship) and that's exactly where the "lost"
energy went in this puzzle. In the spaceship's rest frame the recoil
carries no energy (in the limit as dM << M) but in the observer's frame,
where the spaceship is already moving, the recoil does carry energy.

More specifically and intuitively, the recoil, in the observer's frame,
isn't "3-space parallel" to the path of the photon, and a component of the
recoil lies along the spaceship's path. That's the component that adds
energy to the ship.

For a longer treatment than you probably want to wade through see
message ID: <pan.2006.02.02.15.53.41.359964@xxxxxxxxxx>
elsewhere in this thread.



PD

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