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



In the context of:
[...] I have one box B containing the two photons in
your example and I subdivide B into two parts which I call box G and H
such that one of the photons is in G and the other is in H, then, at
any one instant in time, there is no mass in box G and there
is no mass in box H but there is mass 2q in box B? [1]

On 10/01/2015 05:14 PM, Jeffrey Schnick asked:

Where within B is the mass?

Simple question, complicated answer. Executive summary: If
you have a black box and ask "where is the mass inside", the
laws of physics do not guarantee that you will get an answer.
You can do experiments, but the experiments don't necessarily
tell you everything you want to know.
++ You can measure the zeroth moment of the mass distribution
(i.e. total mass) easily enough.
++ You can measure the first moment easily enough, and hence
obtain the CM position.
-- Higher moments are much tricker to obtain, and might well
exhibit time-dependent monkey business.

In more detail:

1) If we take the question literally, it doesn't have an answer
and doesn't need an answer.

There are lots of questions in this general category. For
example, you could ask, Where is the beauty in this page of
sheet music? Is it in this quarter-note? Is it in this
dotted-eighth-note? When you have something that is extensive
and conserved, you might get to ask where it resides ...
otherwise not.

2) Actually, even if something is extensive and conserved,
you don't necessarily get to ask where it is. Given a single
atom in a specified energy eigenstate, you don't get to
ask where the electron is, or where the charge is. Given
an /ensemble/ of identically-prepared atoms, you can ask
about the probability density, but that's the best you
can do.

3) Even when something is conserved, extensive, and non-
quantum-mechanical you might have a hard time locating "the"
mass based on macroscopic black-box observations. Consider
for example a boxcar containing a troop of hyperkinetic
monkeys. As the monkeys jump around, the CM of the box
remains unchanged, but the details of the mass distribution
are constantly changing.

If we think of mass as "resistance to acceleration", that
only works if the whole system is uniformly accelerated.
If you try to accelerate just the walls of the box without
regard to what the monkeys are doing, you get a result that
is irreproducible and irrelevant.

4) Perhaps closer to the intent of the question, and in any
case to anticipate the obvious follow-up question, let's
consider an experiment that could be done in an attempt
to locate "the" mass.

Consider the gravitational field near the box. For weak
fields, the field equations are linear, so the field due
to the whole box is equal to the field due to the left
half plus the field due to the right half.

At this point we rediscover something interesting about
general relativity: The source term on the RHS of the field
equation is not simply the mass! Each photon in our box,
even though it is massless, is a source for the gravitational
field.

This should be obvious from the fact that a photon passing
near the sun is deflected by the sun's gravitational field.
The photon undergoes a change in momentum. In accordance
with the third law, the sun must undergo an equal-and-
opposite change in momentum, which we explain in terms of
the gravitational field of the photon.

So the field adds up as it should. The experiment gives us
a reasonable answer ... but does not tell us where the mass
is.

For source objects that aren't photons, by far the dominant
contribution to the gravitational field comes from the mass,
so Newton was very nearly right ... but not exactly.

5) If you want simpler behavior you can get it, by a suitable
construction. In particular, an optical /standing wave/ that
fills the box corresponds to a uniform distribution of mass
density.



To summarize: In general, given a box, you don't necessarily
know how the mass is distributed inside.
++ You can measure the zeroth moment of the mass distribution
(i.e. total mass) easily enough.
++ You can measure the first moment easily enough, and hence
obtain the CM position.
-- Higher moments are much tricker to obtain, and might well
exhibit time-dependent monkey business.