Re: A small conservation of energy puzzle

Here's a slight clarification, which I should have included earlier.

This is just a "relativistic rocket", which is a classic problem in
intro physics texts. However, the exhaust angle is funky, and that's
what's confusing.

There's a spaceship of mass M, _plus_ fuel of mass dM.

In the ship's rest frame, when it spits out a photon, the energy of the
ship+fuel changes like this:

M + dM ---> M

The _system_ "ship+fuel" loses energy in the ship's rest frame; that
energy turn into the photon. However, the portion consisting of the ship
alone, of mass "M" (excluding the fuel), neither gains nor loses energy.

In the observer's frame, the energy of the ship _plus_ the fuel changes
like this when it emits a photon:

(M+dM) * gamma ---> M * gamma'

The ship+fuel system certainly loses energy.

However, if we look _just_ at the piece of rest mass "M" -- the ship
without the fuel -- its energy changes like this:

M * gamma ---> M * gamma'

and it gains energy, because the ship sped up, and gamma' > gamma


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