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Re: [Phys-l] Invariant mass and relativist mass...



On 02/25/2008 12:59 PM, Rauber, Joel wrote:

My one sentence version of the difference is as follows:

Do you prefer to write your relativistically correct momentum formula
as:


a) p = gamma*m*v


or



b) p = m*v



If you like (b) you are preferring a relativistic mass approach.

If you like (a) you are preferring an invariant mass approach.

That's true as written, but there's no physics in the distinction, as
discussed below.

______________________________________________________________________

OTOH, there probably is some reason why most current GR and SR (general
relativity and special relativity) practitioners follow approach (a),
and its not simply a matter of faddism.

Yes, there is a reason. Several reasons in fact.

The reasoning starts by recognizing that invariant mass is only
part of the modern (post-1908) understanding of relativity. We
also have valuable notions of invariant length and invariant time.

A sufficient reason IMHO is pedagogical: One approach to teaching
relativity emphasizes that it is weird and rife with paradoxes,
using rulers that can't be trusted and clocks that can't be trusted.
In contrast, it seems in every way better (pedagogically and practically)
to treat relativity as not particularly weird or paradoxical; indeed
it is about as close to the geometry of ordinary Euclidean space as
it possibly could be. Time and space are most easily understood in
terms of proper distance and proper time, which do not contract or
slow down.

A related reason is that it emphasizes the unity of physics. Working
with 4-vectors not only borrows from what we know about 3-vectors, it
spirals back to reinforce and enrich our understanding of 3-vectors.
Similarly, treating a boost as a rotation in the xt plane reinforces
our understanding of ordinary rotations in the xy plane. Just as a
spacelike rotation of a ruler does not change its length, the unity
of physics suggests that a boost should not change its length, either.
http://www.av8n.com/physics/spacetime-trig.pdf
http://www.av8n.com/physics/odometer.pdf

Perhaps most importantly, according to a modern (post-1908) understanding
of relativity, there is *NO PHYSICS* in the distinction between
equation (a) and equation (b) above. It is purely an accident of
the terminology. This can be seen in the following:

c) p = m*u

where here p is the four-momentum, u is the four-velocity, and m
is the invariant mass. It must be emphasized that although this
is superficially structurally the same as equation (b), it adheres
to the modern spacetime approach, using the invariant mass as in
equation (a). So the difference in structure between (a) and (b)
doesn't really tell us anything useful.

The point here is the
*) what is conventionally called the 3-momentum is the spacelike part
of the 4-momentum, whereas
*) what is conventionally called the 3-velocity is *not* the spacelike
part of the 4-velocity.

To repeat, there is *NO PHYSICS* in this distinction; it is just an
unfortunate accident of the conventional terminology. For details, see
http://www.av8n.com/physics/spacetime-acceleration.htm

This is unfortunate in the sense that it is a major source of confusion
for students.


Last but not least, there is an Occam's Razor argument.
-- Introducing relativistic mass, plus time-dilated clocks, plus
FitzGerald-Lorentz contracted rulers is not a complete description
of relativity. You also need to say something about the breakdown
of simultaneity at a distance. And then you need to worry about
transverse mass versus longitudinal mass versus rest mass. And
so on.
-- In contrast, a modern approach to special relativity is much
simpler. Basically you need to say there are 4 dimensions instead
of 3 (which is trivial), and you need to say that rotations in
the xt plane have a "+" sign in one place where spacelike rotations
would have a "-" sign. That's it. One flipped sign. This gives
you the geometry and trigonometry of spacetime, and essentially
everything else follows from that. I'm not saying you have to derive
everything from simple axioms (although you could). My point is
that Minkowski spacetime is just not very different from Euclidean
space, and this greatly reduces the amount of stuff that needs to
be remembered. By the same token, it reduces the number of ways
that mistakes and so-called "paradoxes" can creep in.