🚀 go-pugleaf

markc@smsc.sony.com (Mark Corscadden) writes:

>No!  Here's a statement of my position that relativistic mass can't be
>dispensed with when dealing with the mass of macroscopic objects (looking
>again to spare helpless straw men from beatings):

>	It's perfectly reasonable to talk about the mass of a
>	macroscopic object.  The inertial mass of the object
>	can be operationally defined in terms of the force
>	that has to be applied to the object to produce a given
>	rate of acceleration. 

In order to spare straw personnel further injury, let me state that
the relativistic mass is obtained in this manner not from F = ma,
but from F = dp/dt and p = mv.  I'm sure you know this, though.
This has no real bearing on the rest of your statements, except
for the additional caveat that to get the rest mass this way,
you have to define quantities in the object's rest frame.  

[a bit deleted] 
>	Yet you can't make this
>	notion of mass work in *full generality* without using
>	the notion of relativistic mass.  More precisely, if
>	you reject relativistic mass then the amount of mass
>	you have on hand is subjective and completely dependent
>	upon how you think the objects on hand should be viewed.

This is quite true.  Mass is no longer a substance you have in various
amounts; you can only define it for a system, having first drawn a
boundary around the system, so to speak.  This seems to be the
primary thing that bothers you and Leigh, and it's made most
apparent in the situation with the cannonballs in the box. 

I don't think that this is that much of a problem, though.  Consider
an analogous situation with three-dimensional vectors.  Suppose you
have a complicated system with a lot of particles flying around.  
The particles are peculiar in that they are constrained to fly at
a constant speed: the magnitudes of their momenta are always
constant, and this magnitude depends in a simple way on the type
of particle.  You would agree with me (since you like the use of
mass to mean rest mass in particle physics) that this magnitude would
be more worthy of a special name than, say, the x-component of
momentum.  Let us borrow a word from R. P. Feynman and call it
"wakalixes."

Now look at the whole collection of flying particles.  How 
much wakalixes is there?  You'll get much more if you add 
up the magnitudes of individual momenta than if you find the magnitude
of the momentum of the whole system.  For a situation in which the
boundaries of the system under consideration are unclear, the magnitude
of the momentum is ill-defined.  The x-component of momentum, on the
other hand, can be treated as a local density that adds quite nicely,
and the dependence on the boundaries of the system is very simple.

This doesn't mean that we have to take the word "wakalixes" and
suddenly apply it to the x-component of momentum when we deal 
with macroscopic objects.  The wakalixes as defined previously is
quite permanent as long as we don't let momentum flow into or
out of the system, and, furthermore, it's rotationally invariant,
which the x-component is not. 

Even in the real world, of course, we don't seem to have much
trouble dealing with the magnitude of a macroscopic system's 
momentum, even though the system may consist of Avogadro's number
of moving parts.  It's dependent on how you group objects together
into systems, but it's a useful quantity, and more significant than
any one component taken alone.
 
Now, this is, of course, a terribly unfair parable, not because
of the ridiculous nature of the assumptions, but because I haven't
taken into account the fact that in our everyday lives we are almost
always dealing with reference frames very, very close to each other.
The real analogy would be with a world like the one described above
in which, in addition, momenta usually pointed in the x direction with
a maximum deviation of a thousandth of a degree.

However, special relativity was formulated to deal with situations
far from the everyday norm, and in my imaginary world the equivalent
theory would deal with how things look like when things go in radical
and strange directions far from the x-axis.  If the full theory possesses
rotational symmetry, when teaching it to students you shouldn't give
them the idea that the symmetry goes away in the macroscopic limit.

I must also repeat Leigh's reminder that this argument is about
words and pedagogy, not physics.  I'm increasingly of the idea that
we should at least mention both definitions to students, so that
they know what various people are talking about, but work
primarily in terms of mass = rest mass.  To understand how the two
definitions differ is to understand a lot about special relativity.
It's okay to add lots of details as long as you make them as
coherent and consistent as possible:  more data makes it easier
to detect a pattern.  

As for the cannonball situation, I don't see any trouble as long as
the experimenter is properly acquainted with special relativity.
Measure the rest masses of all the parts of the system by the
usual procedure (being sure to put yourself in the rest frame of
each flying object first); then find the total 4-momentum of the
system, since you know the relative speeds (after all, you needed
to catch up with the flying cannonballs to do the measurements!),
then use the fact that the mass of the whole system is the invariant
length of this 4-vector.  It sounds quite indirect, but it can be
found without using the idea of relativistic mass.  You might say
that the relativistic mass is one component of the 4-momentum, but
I didn't have to think of it that way.
 
(Also, remember that you have to be careful when tugging on the
sealed box to measure the mass initially:  how the box accelerates
is not necessarily how the system of box plus cannonballs accelerates,
unless you measure average acceleration over a large enough time that
the cannonballs can bounce around many times!  This simple-seeming 
operational definition can have pitfalls of its own.)
 
-- 
Matt McIrvin  mcirvin@husc.harvard.edu