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From: sra@oddjob.UUCP (Scott R. Anderson)
Newsgroups: net.physics
Subject: Re: Definition of mass in relativistic mechanics
Message-ID: <996@oddjob.UUCP>
Date: Mon, 14-Oct-85 01:35:47 EDT
Article-I.D.: oddjob.996
Posted: Mon Oct 14 01:35:47 1985
Date-Received: Tue, 15-Oct-85 09:56:39 EDT
References: <576@bonnie.UUCP> <1997@brl-tgr.ARPA> <581@bonnie.UUCP>
Reply-To: sra@oddjob.UUCP (Scott R. Anderson)
Organization: University of Chicago, Department of Physics
Lines: 59

In article <581@bonnie.UUCP> emh@bonnie.UUCP (Ed Hummel) writes:
>
> Most texts (even recent ones)
> and all the early papers use the result (definition):
> 		mass = gamma * (rest mass).
> This  is consistent with keeping the Newtonian formula for momentum.
> 	Most of the professional physicists that I know, do not use the
> word "mass" according to the above definition, but use it to mean "rest
> mass".  The preferred usage seems to be to redefine momentum:
> 	momentum = gamma * mass * velocity.
> Where mass is understood to be "rest mass".

Given the first definition, this is *not* a redefinition of momentum, just
of the term 'mass'.  Momentum is the important quantity here, as it is the
momentum which changes with the application of an external force.

> Of course, the same goes for energy:
> 	E=m*c**2	->	E=gamma*m*c**2.
> I would like to hear opinions about:
> 	What usage is more common?
> 	Are there good reasons for preferring one definition over the other?

It all depends on the situation in which it is being used.  In the study
of relativity, it is sometimes useful to consider the properties of a
relativistic particle in your (rest) frame of reference.  The noticeable
property here is that the particle becomes more difficult to accelerate
as its speed increases.  In this case, it is useful to think of the 'mass'
as gamma * (rest mass), because this fits in with our Newtonian idea that
a more massive particle is harder to accelerate.  This is therefore the
way that relativity is usually taught, to base it in one's previous
training in classical mechanics.

However, particle physicists by necessity work with *two* frames of reference
on a regular basis, the 'lab' frame and the 'center of momentum' frame.
The latter is considerably easier to work with since the total momentum is
zero, but it is necessary to work with the former because that is where the
experiments take place.  PP's are therefore constantly shifting back and
forth between these two frames.  It is therefore most useful to work with
invariant quantities, in particular the four-momentum, (E, px, py, pz).
The square of the rest mass is the 'length' of this four-vector,

		m^2 = E^2 - (px^2 + py^2 + pz^2)

and is therefore also an invariant under the Lorentz transformation.

One other thing to keep in mind is that, because c is an invariant, the
energy E = gamma * m * c^2 is equivalent to the non-invariant mass "gamma * m".
Why use two names for the same thing?  Besides, if you use the right set of
units, c = 1, there is no difference at all! :-).

>Take temperature, for example.  It is a well defined concept.  It is
>not an invariant.

Temperature is essentially the total energy of a system; that is why
'temperatures' in the early universe are quoted in energy units such as
TeV.  Of course, this is in the CM system of the universe.

				Scott Anderson
				ihnp4!oddjob!kaos!sra