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Re: [Phys-L] foundations of physics: Galilean relativity



On 10/03/2015 08:50 AM, Moses Fayngold wrote:

It is important to note that condition of being at rest in the
system's CM is not enough to determine its rest energy.

Actually it's not important. Being at rest relative to the
CM(*) is /more/ than enough.

If the parts of the system interact with one another, then the
distance between them is very important.

No, it's not important. Re-arranging the energy and momentum
/within/ the system has no effect on the [energy,momentum]
4-vector of the system as a whole, and therefore no effect
on the mass. The proof of this is about three lines long.
1) Assert that [E,p] is conserved.
2) Assert that the dot product is a Lorentz invariant.
3) Therefore system mass is constant (for any closed system)
and Lorentz invariant.

The field energy of e-p pair is much greater when the members of the
pair are widely separated than when brought to rest close together.

The field energy is part of the system energy.

By bringing them close together while still keeping them at rest
(call this process "adiabatic contraction") we can make their net EM
field negligible.

That process requires transferring energy across the boundary
of the system, or storing energy in some as-yet-unmentioned
part of the system. Either way, looking at the pair does not
account for all of the /system/ energy.

To repeat: In /any/ frame (not just the rest frame) you can
calculate -p•u, where p is the 4-momentum and u is the 4-velocity
of the center of mass(*) of the system. I claim that -p•u is the
rest energy of the system. In the rest frame this is obvious,
since we know the components of u in that frame. If you choose
some other frame -- or no frame at all -- it may be less obvious,
but it is no less true: -p•u is the rest energy.

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


(*) Nitpickers note: Obviously this whole discussion is restricted
to systems that have nonzero mass. Otherwise:
a) The system has zero mass.
b) The system doesn't have a rest frame or a CM.
c) Determining the mass does not require using the rest
frame, or any other frame, since we knew the answer
before we asked the question.