Re: [Phys-L] Definition of Mass

[John Denker <jsd@av8n.com>]

On 10/23/2013 10:15 AM, coachpayne@aol.com wrote:
> Interesting statement from the paper by Gregg Swackhamerin the
> Modeling Resource paper CognitiveResources for Understanding Energy
> (Draft March 31, 2005): “since over 90%of the mass of Mt. Rainier or
> bread, arises not from particles but from energystored in the strong
> fields holding the constituent quarks together, it is apretty
> accurate statement in many instances to say that energy is there
> justlike bread in a breadbox.” (Wilczek, Frank, “Mass without Mass I:
> Most ofMatter,” Physics Today, November, 1999, pp. 11–13) This might
> be commonknowledge to most or all of the folks on this list, but it
> was novel to me. I’denjoy reading your comments on this.  Howdo
> others of you define the term mass?  Iunderstand the inertial and
> gravitational meanings, but how do you discuss massin your classes?

There are several different ideas mixed in there.  Let me
take a stab at them, in no particular order.

First of all, the paper by Wilczek is excellent:
  http://scitation.aip.org/content/aip/magazine/physicstoday/article/52/11/10.1063/1.882879
  http://ctpweb.lns.mit.edu/physics_today/phystoday/MassI.pdf

Wilczek doesn't talk about "bread in a breadbox" and I probably
wouldn't either.  I'm not sure students would find that expression 
particularly informative or helpful.

As for what I would say in class, that depends quite a lot
on what class it is ... and how far along we've gotten.  The 
question that was asked is outside the scope of the typical
introductory course.  It would be more at home in a second-
year course for physics majors, perhaps E&M or modern physics.

It is kinda comical to see introductory textbooks making a big
fuss about ``wave/particle duality'' and a big fuss about the
distinction between ``matter'' and ``interactions'' when modern
physics has long since turned its back on all such notions.  As
Wilczek explains, basically, there's just fields.  The electron 
is an excitation of the electron field, just as the photon is 
an excitation of the electromagnetic field, and the same goes 
for quarks and gluons and everything else.  The field equations 
are not exactly the same in all cases, but the similarities are 
more important than the differences.

If this sounds complicated, you're thinking about it the wrong
way.  Actually it's a great simplification.  There's only one
kind of stuff.  That's simpler than having lots of different
kinds of stuff.

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

As for the definition of mass, there are many different notions
of mass ... all of which turn out to be equivalent, if you look
carefully enough.  Therefore you can pick any one of them as 
"the" definition, and then turn the crank to show how one
meaning connects to the other.

YMMV, but my personal preference -- in the advanced course -- 
would be to emphasize that mass is the invariant norm of the 
[Energy, momentum] 4-vector.  The four-vector has to have
"some" norm, and we choose to call it the mass.  That is:

  m^2 c^4 = E^2 − ps^2 c^2              [1]

> From there it is easy to convince yourself that mass is the
constant of proportionality that connects momentum to velocity.

Mass is also the constant of proportionality that connects
energy to the square of velocity, correct to second order.

Mass is also the constant of proportionality in the famous
expression E = m c^2 ... which is a zeroth-order approximation
to equation [1].

And so on.  It is one of the glories of special relativity that
it lets us see all this as one fact, rather than N little facts 
that would have to be learned separately.  You start with 
equation [1] and then pick out the zeroth-order piece, the
first-order piece, and the second-order piece.

For details, see  http://www.av8n.com/physics/mass.htm

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

As for the equivalence of inertial mass and gravitational mass,
you have to be careful how you express it.  Modern theory, as
confirmed by experiment, tells us that strictly speaking,  mass 
is /not/ the only source-term that gives rise to the gravitational
field.

Equivalence holds locally, for a nice uniform field with sources
that are not moving too fast.  In other words:  It only tells us 
about the classical limit.  This is, however, super-important to
the theory.  Via the correspondence principle, this pins down a
lot of things (but not everything) in the theory.

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

As for the question of "where is the mass" -- in the "particles"
or in the fields -- that's quite a tricky question.  The Wilczek 
article oversimplifies this a little bit.

> energy stored in the strong
> fields holding the constituent quarks together

That is another statement that does not come directly from
Wilczek.  It is in some sense diametrically wrong.  In fact,
the binding energy does not explain the mass or even add to
the mass;  in some sense it /subtracts/ from the mass.

You can see this already in the periodic table.  The mass per
nucleon of something that is tightly bound, such as an iron
nucleus, is *less* than the energy of an isolated proton or
neutron.  There's a term for this:  mass deficit.  The binding
energy corresponds to a mass deficit.  This makes sense in
terms of E = m c^2, because to unbind the iron nucleus you
have to do work on it, i.e. you have to add energy to the
system.

So it would be better to say that the field contributes to
the mass *OR* to the binding, one or the other, not both.
The more binding the less mass.

When you say that in words it might sound backwards and confusing,
so it really helps to draw the energy-level diagram.  Binding
energy is measured /downwards/ from the top, whereas the energy 
of an energy level is measured upwards from the bottom.
  http://www.av8n.com/physics/mass.htm#sec-binding-ke

A useful concept that goes with this is /dressed states/.
You can see the phenomenon already in classical fluid 
dynamics:  If you drag an object under water, even if
you ignore friction you find that the object has more mass
than it would in air.  That's because when the object is in
motion, the water must also be in motion to make way for the
object, and the KE of the water contributes to the KE of
the /dressed/ object (whereas the mass of the "bare" object, 
by definition, does not include this contribution).

Using this fancy modern language, the bare mass of many of 
these so-called particles is very small, possibly zero.
The experimentally-observed mass is the dressed mass, and 
includes contributions from dragging around bits of various 
other fields.

Another useful concept is /screening/.  You can see this
already in the electrostatic force between two charged
objects in a dielectric medium.  Examples include H+ and
OH- atoms in aqueous solution, where the electric field is
screened by the dielectric constant of the water.  Another
familiar example is electrons and holes in a semiconductor.
We can apply this to the less-familiar situation of nucleons
(or quarks) in a nucleu.  The the strong force is heavily
screened, and this affects the mass.