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

["Ken Caviness" <caviness@southern.edu>]

If you like to look at it that way.

I'm not actually claiming that the idea of "relativistic mass" is an
error.  I see some redeeming qualities to the concept, particularly in a
class that considers implications of physics theories on our worldviews.

Consider:  Relativity shows us that treating space & time as having
separate existence is not particularly useful.  There is something we
call "spacetime" that has space & time projections or components.  A
spacetime interval can register for different observers as being made up
of varying amounts of space and time, in a way somewhat comparable to
distance in the plane, which can be viewed as a change in x combined
with a change in y:  observers having different orientations for their
x- & y-axes disagree on the Delta-x & Delta-y values, but agree on the
distance = sqrt((delta-x)^2 + (delta-y)^2)^1/2.  For some uses, the x- &
y-coordinates are useful, but this lends credence to the view that the
spatial distance is more fundamental (invariant under rotations,
reflections and translations).  The spacetime interval is invariant
under the Lorenz transformation: it's the same for all inertial
observers in special relativity, while distance and time intervals will
vary according to the observer.  So: time and space (time periods and
space distances) separately aren't as fundamental in special relativity.
So what else depends on the observer?  Is "everything relative"?  No,
any particular spacetime interval is the same for all observers, the
speed of light in vacuum is measured to be the same by all observers.
And the laws of physics (or better: the equations we have found that
interrelate various physical variables) can be expressed in a coordinate
independent form, valid for all observers, although the values of the
position, length, velocity, acceleration, force, etc., are not
invariant.  So we come to "F=ma" (appropriate vector indications
assumed):  It's natural to ask whether the equation is already in a form
valid for all observers and which if any of the variables linked are
invariant under the Lorentz transformation, which are
observer-dependent.  "F" & "a" are observer-dependent, and not in the
same way.  I see relativistic mass as an attempt to save F=ma while
conceding that m varies for different observers.  "F = gamma m a"
attempts to put the equation in a more generally valid form, and
identifies the rest mass "m" as an invariant.  Similarly for the more
general "F=dp/dt" with "p=mv".  It was thought that by conceding the
observer-dependence of m (not too big a stretch, if space & time depend
on the observer), the formulas could be applied as is.  If mass is
treated as an invariant, we must modify the definition of momentum to "p
= gamma m v" and admit the old version to be an approximation.  So far
either view could be seen as justifiable.

Advantage of relativistic mass:  It's easy to combine with the idea of
mass increasing with clocks running slower and lengths contracting as
one's speed increases.  Simply stated, if you push at a constant rate
the resulting acceleration does _not_ remain constant as the simple F=ma
would make us think, but a decreases "as if" the mass were increasing.
It makes it easy to accept that I'll never be able to reach the speed of
light.  Of course you have to explain that the "moving observer" sees
none of this:  she is at rest in her own frame of reference -- no time
dilation, etc.  

Advantages of using invariant (rest) mass everywhere:  Just as the
spacetime interval is the invariant constructed from distances & time
periods by ds^2 = dr^2 - (c dt)^2, (m c^2)^2 = E^2 - (p c)^2.  It is
common to use units where c=1 and write ds^2 = dr^2 - dt^2, m^2 = E^2 -
p^2.  This parallel reveals (rest) mass as the invariant formed by
energy & momentum:  very nice!  Also, as I tried to (briefly!) say
earlier, the concept of a single quantity called the relativistic mass
(gamma m) just cannot be used as generally as mass can in Newtonian
physics.  Imagine an object having a different effective mass depending
on which way I push!  We can decide to use this system but it is awkward
and becomes less intuitive the farther we take it.

So my vote remains "anti":  little mention of "relativistic mass",
consistent use of rest mass "m", emphasis on the gamma factor arising
from dtau/dt, so as we take further derivatives it's unsurprising when
we have more gammas showing up.

Enjoy,

Ken


> -----Original Message-----
> From: Tom Sandin [mailto:sandint@ncat.edu]
> Sent: Wednesday, February 27, 2008 2:28 PM
> To: Forum for Physics Educators
> Cc: Ken Caviness
> Subject: Re: [Phys-l] Invariant mass and relativist mass...
>
> At 12:39 PM -0500 2/27/08, Ken Caviness wrote:
> > To summarize,
> >
> > When force and velocity are perpendicular, F = gamma m a.
> > When force and velocity are parallel,      F = gamma^3 m a.
> >
> > The product (gamma^3 m) was at one time referred to as the
> "longitudinal
> > mass", as distinguished from (gamma m), the "transverse mass".  To me
> > the non-existence of one simple "relativistic mass" covering all
cases
> > is a good reason to join (or stay in) the "anti" camp.
>
> If you start out with an error (there are two kinds of relativistic
> mass), you arrive at an erroneous conclusion ("the non-existence of
> one simple "relativistic mass""). Correct?
>
> Tom Sandin