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

[John Denker <jsd@av8n.com>]

On 10/01/2015 09:40 AM, Jeffrey Schnick wrote:

> I have been thinking in terms of a control mass system [...] Your 
> example is expressed in terms of a control volume system

I suppose that's true, but I don't think it changes the physics at 
all.  It seems to me, you can define "+" to mean adding particles
or adding volumes, and it doesn't matter in this context, because
as soon as you write

	system G "+" system H = system B

then you can have a situation where G and H are each massless,
but B is not.  I realize that everybody had "conservation of mass"
drilled into their heads in high-school chemistry, but Mother
Nature has other ideas.

It's a simple calculation.  Just add the components.  Given two 
photons G and H that we group together to make B, the four-momenta
are:

  G	 =	 [q, +q, 0, 0]@lab
  H	 =	 [q, −q, 0, 0]@lab
  B	 =	 [2q, 0, 0, 0]@lab

Then calculate the mass.  For diagram and details, see
  https://www.av8n.com/physics/spacetime-welcome.htm#sec-invariance-conservation

[scenario snipped]

> You thought this left 0 mass inside the box because the only thing
> in the box was a photon and it has zero mass.

Yes, I thought exactly that, and still think that.  If the only
thing in the box is the one running-wave photon, then I know
the [energy,momentum] 4-vector and I can calculate the mass.
I still think it's zero.  It's an easy calculation.

> What you didn't notice was that amount q of system mass was still
> inside the box.

Where's That From?  It seems to rely on some intuitive notion
of "system mass" that I don't know how to calculate.  Given the
choice between that and the plain old mass that I do know how
to calculate, I'm gonna stick with the latter.  It is easy to 
keep track of the energy and momentum, because they are conserved 
quantities ... and then it is easy to use them to calculate the 
mass.  I doubt there is a 4-vector anywhere that corresponds to
"an amount q" of mass inside the box.  We have only two photons
to play with, and it's pretty much a binary choice as to whether
you group them together or don't.
  https://www.av8n.com/physics/spacetime-welcome.htm#sec-invariance-conservation


> The locality of the conservation of anything is best expressed
> (perhaps can only be expressed) in terms of a control volume system
> so I want to go there.

If we exclude continuous fields and consider only discrete
particles, as we've been doing here, I reckon the conservation
law can be expressed either way (control mass or control
volume).   Conservation becomes a statement about continuity
of world lines.  For diagrams and discussion, see:
  https://www.av8n.com/physics/conservation-continuity.htm

Conservation of charge corresponds to continuity of electric current,
which corresponds to continuity of the world line for an abstract
quantum of charge (not to be confused with any particular charge-
carrying particle).

  Note that electric current is *not* a conserved quantity.
  We have conservation of charge, but /continuity/ of current.

In the introductory course, the particle-based formulation is
easier to formalize.  After all, F=ma is a particle-based concept.  
The control-volume formulation is easier to /visualize/ ... but 
to quantify it in a useful way, in a way that students can connect
to other stuff they know, is a bit of an uphill slog.  I suggest
starting with the special case of electric charge, where we have 
well-behaved ammeters to measure the current, and generalizing 
from there.