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Re: [Phys-L] momentum conservation, particle annihilation, different reference frames



There are about six things going on in this scenario.
Here is a new guess about which one is causing the
confusion. We start with a parable:

Inside a moving spaceship, suppose we have a pair of
baseballs floating around loose. The pair is held
together by a string and pushed apart by a spring.
At time t=0 we burn the string, and the baseballs
pop apart. The direction doesn't matter much.

Eventually each baseball hits a bulkhead. There
is chewing gum conveniently located there, so they
get stuck.

We are now going to claim that when the pair
was floating around loose, it was not "part of"
the ship ... but now that they are stuck, they
are part of the ship. Whatever momentum they had
(in the lab frame) has now been transferred to the
ship.

Does this added momentum cause the ship to speed
up? No, because the mass of the ship is now larger.
The larger momentum in the numerator is canceled
by a larger mass in the denominator, and the velocity
of the center-of-mass stays strictly constant.

Of course various parts of the ship can move
relative to the CM, like a pigeon bobbing its
head, but we assume these motions are bounded.
The velocity of each part of the ship tracks
the CM, plus or minus a little bit, and the
discrepancies average out over time.

=============

You can play the same game with fuel particles
and photons.

Whatever momentum (in the lab frame) was originally
in the fuel particles eventually gets dumped into
the fuselage of the ship. If we claim the fuel
was not originally "part of" the ship, then some
momentum has been transferred and is now part of
the ship. This is mostly a word game, but it's
not wrong.

However, some mass has also been transferred and
is now part of the ship. The velocity of the CM
is strictly constant. The velocity of various
subsystems will track the CM on average.

Don't forget the timelike component of the 4-momentum.
The photon carries energy as well as momentum. When
the photon is absorbed, the mass of the absorber
goes up ... even though the photon was massless.

Again: You have to get past the HS chemistry idea
that mass is conserved. All four components of
the momentum vector are conserved, and the norm
thereof is the mass ... but the norm is not
conserved! The norm is a nonlinear function
of the four components. (If it were linear it
would be conserved, but it's not.)

This is one of the eleventeen ways that you know
that energy and momentum have to be part of the
same 4-vector. If they weren't, you'd be able to
do creepy things like speed up the CM by rearranging
things inside the spaceship. Since the 3-momentum
depends on the reference frame, it's hard to imagine
being able to monkey with it without violating
Galileo's principle of relativity.