🚀 go-pugleaf

It seems that the "relativistic mass" thread has come to a happy
conclusion.  At first the suggestion of doing away altogether with
the concept of relativistic mass seemed to me to have problems.
Lorentz invariant mass was described by several people who were
explaining how my concerns are dealt with, which made me pretty
sure that my concerns were not real problems.  I still felt uncertain
as to whether or not I really understood exactly what was being said
concerning invariant mass, though.

I isolated the question about invariant mass that was still causing me
doubt, presented it as a simple example, and asked whether the answer
I came up with was the right one.  Basically I got a number of replies
saying "yes, that's pretty much correct".  Very good!  I want to thank
Matt McIrvin, David Knapp, and Matthew Austern for checking my new-
found understanding of Lorentz invariant mass and also thank John Baez
for helping me get to the point where I could do real calculations.

These thank-you's were the important thing.  I think the rest of this
article will not be of interest to someone who has studied Relativity
in detail, but laypersons (like myself) who are in doubt about the
feasibility of doing away with "relativistic mass" might find it
interesting.  All I'm doing is just reviewing the doubts that troubled
me (and might trouble other non-specialists) and then describing how
the understanding that folks have been sharing with me answered those
doubts.

The root of the problem seems to be in how I was thinking about mass
and energy.  Take the rest mass of a block of material for example.
When it's heated the rest mass increases.  Now the way I thought about
this was:

	You heat the material.  The molecules move faster,
	which means that the (relativistic) mass of each
	molecule increases.  That's how relativistic mass
	works: the faster something is moving the more mass
	it has.  So the total rest mass of the block, which
	is the sum of the (relativistic) masses of all the
	molecules, increases.

Then I heard people saying that the mass of an object does not increase
as it moves faster, that this is the idea of relativistic mass and it's
an idea that just causes confusion and should be avoided.  The mass of
an object doesn't change just because it is moving with respect to you:
this is the modern viewpoint.

My problem was that this completely invalidated my original image (described
above) of how the additional motion due to heating a block of material
increases the block's rest mass by increasing the (relativistic) mass of
each individual molecule.

This is how I would now answer the concerns that I originally had (and
that other non-specialists might also have):

	It's not that anyone is saying that the mass which *you*
	think of as the "additional relativistic mass of each
	molecule" doesn't exist.  What they're saying is that
	the mass is there but that you're *thinking* about it
	wrong.  Don't think of it as additional relativistic mass
	which belongs to each individual molecule.

In other words, rejecting relativistic mass doesn't mean that you
don't get the same value for the mass of a block of heated material,
it just means that you get that value using different, better tools.

Okay, the tools for computing the total rest mass of a composite object
using the idea of relativistic mass (the old, bad way) are:

1)  Find a reference frame where the composite object is more-or-less
    at rest.  Don't ask how, this is the bad old way so you shouldn't
    be surprised to find out that some questions don't have very good
    answers.

2)  Determine how much (relativistic) mass each molecule has.  The
    faster the molecule is moving the more mass it has, you know.

3)  Add up the individual masses to get the total.  In this case, if
    you just focus on the change in mass due to heating the material,
    relativistic mass is additive so this works out.

So if that is the old bad way, what is the new good way?  (Disclaimer:
innocent laypersons should not expect a 100% trustworthy answer from me.
I think the following is correct, but I'm not a professional and just
started learning about this stuff myself so consider yourself warned.)

The good way to compute the rest mass of a block of heated material:

1)  Pick a reference frame, any frame!  It doesn't matter whether the
    block of material is more-or-less at rest in that frame or moving
    at 0.99 lightspeed.  So much for one problem that you have using
    the "bad old way" that disappears when you start using Lorentz
    invariant mass.  The invariant mass is the same in all reference
    frames so you don't have to determine before-the-fact which frame
    seems to be the rest frame for this object.

2)  The invariant mass of each individual molecule doesn't change as the
    material is heated.

3)  The invariant mass of the block of material does increase as it is
    heated.

4)  From (2) and (3) you can see that the mass of a system is not equal
    to the sum of the masses of the constituent parts; invariant mass
    is not additive.  This is not a problem, it's just the way that
    (invariant) mass works.

5)  You need two neat new quantities in order to compute the rest mass
    of the block: total energy and momentum.  (Everything is measured
    in units such that lightspeed is 1.)  Let

	gamma = 1/sqrt(1 - v dot v)

    where v is the velocity vector of a specific molecule and "dot"
    is the vector dot product.  For those who haven't messed with dot
    products, v dot v is just the square of the speed, where speed
    is just the magnitude of the velocity, ignoring its direction.

6)  The total energy of an individual molecule is

	E = m * gamma

    where m is the invariant mass of the molecule.  Note that E is
    not an invariant quantity, it's different when measured in different
    frames.

7)  The momentum of an individual molecule is

	p = m * gamma * v.

    Momentum is also a frame-dependent quantity.  Note that p is a vector,
    whereas E is a quantity who's value can be expressed with a single
    number.

8)  Now you compute E and p for the whole block of material by adding
    up the values of E and p for the individual molecules.  That's
    because E and p *are* additive!

9)  Finally, you compute the invariant mass of the block of material
    using the formula:

	m = sqrt( E^2 - (p dot p) ).

    This value is independent of the choice of reference frame which you
    used when measuring the values of E and p for each individual molecule.

I found it fun to calculate with some simple 2-3 "molecule" systems to
see how this works out.  Here is what happens conceptually, at least so
it seems to me from having done some specific calculations:

When you use a reference frame in which the system is more-or-less at
rest, the individual values for E, which are simple magnitudes, add up
in a direct way.  But the values of p, which are vectors, tend to point
every which way and cancel each other out when you add them up using
vector addition.  In the equation

	m = sqrt( E^2 - (p dot p) ),

as you heat the object, the total value of E grows but the value of
(p dot p) stays near zero even though the value of (p dot p) for each
individual molecule gets larger.  Put in a sloppy way, as long as the
momentum vectors of the individual parts of a system are pointed in random
directions they cancel each other out, so (p dot p) is small and doesn't
subtract from the final value of m.  All of the increase in E appears as
an increase in the invariant mass of the system.

When you use a reference frame in which the system has a large overall
velocity in one direction then the momentum vectors add up in the net
direction of motion, except for individual random deviations from
the norm which again tend to cancel out.  Thus the larger value of E
due to this net motion in a single direction is canceled out by the
quantity (p dot p).  Put figuratively, this formula for the invariant
mass very nicely throws out energy that is an artifact of an object's
overall motion but keeps the energy that is locked up in the relative
motion of the parts of the system.

Disclaimer #2: These are of course just my impressions and are subject
to being improved based upon further guidance, if even more guidance
happens to come my way.  Much thanks to the people who helped me see
these issues a little more clearly; any confusion which remains is my
fault, not theirs.

Erratum:  In a previous post I gave the invariant mass of a system
consisting of two 1kg objects moving away from each other at 0.995
lightspeed as 20kg.  Now that I have some reliable tools at my
disposal I've recalculated this correctly.  The answer should have been

	2 * (gamma + 1)

where gamma = 1 / sqrt(0.009975).  That comes to about 22.025, not 20.

Mark Corscadden
markc@smsc.sony.com
work: (408)944-4086