From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Converting the masses
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Mon, 27 Jul 92 22:13:50 GMT

NOTE: John C. Baez is the author of this article.  I am only using my
account to post this article for John as a favor - Mark Corscadden

In article <1992Jul25.194550.1970@smsc.sony.com> markc@smsc.sony.com (Mark Corscadden) writes:

>By the way, this is completely different from the situation for those
>of us who talk freely about relativistic mass.  There the mass of an
>object depends upon it's velocity relative to you, but relative velocity
>is objective; it has a simple definition and can be measured. 

The only thing is, one would be more correct in calling this quantity
energy/c^2.  If you're a mathematical physicist then c = 1 so we're just
talking about energy.  If you want to call energy "relativistic mass"
that's fine with me -- I'll be able to talk faster than you, though.  :-)

>How do you
>"measure" whether it's correct to view several objects as bound together
>in a system or as separate entities?  It's just a subjective viewpoint in
>general, and has the potential of becoming arbitrarily complex to judge.

Quite true.  

>I'll agree that using the sum of the constituent (rest) masses of a
>system gives you a very silly definition of "mass".  But I'd challenge
>anyone to give a better definition of the "mass" of a macroscopic object,
>with no reference to relativistic mass, and then explain how to apply
>their definition in the following pretty simple situation:
>
>	You have a very strong box with walls that are very
>	elastic.  Inside you have two very elastic cannon balls
>	held together by a latch.  Between the cannon balls is
>	a very very strong compressed spring.  (Too strong to be
>	possible for a real spring.)  You can't see inside the
>	box and *don't have to* in order to measure its rest mass;
>	just apply a force to the box and measure the resulting
>	acceleration.  Better definitions for the "rest mass" of
>	this object are welcome, but lacking an alternative this
>	is the definition used here.
>
>	At some time T=0 a little mechanism within the box
>	frees the latch that holds the two cannon balls together.
>	This spring is *real* strong; the two cannon balls are
>	propelled away from each other with a relative velocity
>	of 0.99 lightspeed.  They proceed to bounce around inside
>	the box, which is a very strong box and doesn't break.
>
>	You can still measure the mass of this object after the
>	latch has been freed, and you will find that it hasn't
>	changed.  So open the box.  The cannon balls go flying
>	off in opposite directions at equal speeds (roughly).
>
>How much mass do you have now?  

We all agree on what energy and momentum are in a given inertial frame,
I hope.  (In classical mechanics, these are the generators of time and
space translations respectively.  Ditto for quantum mechanics only now
"generator" means "self-adjoint generator of the unitary group...")  I
would prefer to define mass (in the context of special, NOT general
relativity) to be the quantity m given by m^2 = E^2 - p^2 (setting c =
1).  This has the following properties:

a) It is independent of the choice of inertial frame.

b) In a frame in which E is minimized, the mass is just equal to E.  
We can call such a frame a "rest frame" since in this frame p = 0.  

This allows us to define the mass of any system for which we can define
its (total) energy and momentum, and this definition agrees with the
usual one in the case of elementary point particles.  It is true
according to this definition that a hot chunk of iron has more mass than
a cold one.  

I will not get drawn into arguments over whether this definition is
"right".  As a mathematican who likes to work with sharp concepts, I
would have trouble with any other definition (except the one in which
"mass" is just another name for "energy", in which case why bother
calling it mass?).

I will note that any group representation theorist would love my
definition of mass since it simply says that mass is a central element
in the universal enveloping algebra of the Poincare group.  (Rather
sloppily speaking, a "Casimir".)  This approach also makes clear the
relation between mass in classical and special-relativistic physics: in
the limit c -> infinity, the Poincare group "contracts" to the Galilei
group; the mass appears as a cocycle in the representation theory of the
Galilei group -- see Guillemin and Sternberg's "Symplectic Techniques in
PHysics" for this stuff.  

John Baez
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