Re: [Phys-l] momentum first and relativistic mass

[John Denker <jsd@av8n.com>]

On 02/27/2008 01:19 PM, Rick Tarara wrote:

> Thus, a first pass through SR with slow clocks etc. seems more reasonable to 
> me than immediately jumping into a more mathematical approach.


What's the evidence that the spacetime approach is "more mathematical"?

I see it as less mathematical, more pictorial, more analogical, more
physical.

If you go with "slow clocks etc." you need
 -- slow clocks
 -- shrunken rulers
 -- funny mass
 -- another funny mass
 -- something to describe breakdown of simultaneity at a distance

... and by the time you formalize all of that, it's way more mathematical
than formalizing the structure of spacetime.  Minkowski space has *one* 
funny minus sign in the definition of the dot product, and that's it. 
That's it!  It couldn't possibly be simpler (without being trivial).
And it lends itself to drawing spacetime diagrams, which are helpful
for making all sorts of "paradoxes" disappear.

On 02/27/2008 10:39 AM, Ken Caviness wrote:
> > To summarize, 
> >
> > When force and velocity are perpendicular, F = gamma m a.   [1]
> > When force and velocity are parallel,      F = gamma^3 m a. [2]

Hmmm.  When I do the calculation, being careful to compare apples to
apples, I get
  a) For straight-line acceleration, the 3-acceleration is smaller
   than the spacelike part of the 4-acceleration by 3 factors of gamma,
   in agreement with equation [2] above.
  b) For circular motion, the 3-acceleration is smaller than the
   spacelike part of the 4-acceleration by *two* factors of gamma,
   which seems to conflict with the single factor of gamma in
   equation [1] above.

I just now wrote up the calculation for the circular case and
put it at
 http://www.av8n.com/physics/spacetime-acceleration.htm#sec-circular

Is this a standard result?  It's implicit in the analysis of relativistic
cyclotron motion, which is in lots of textbooks.  It would be good if 
somebody would check my arithmetic.  It should be easy to check.  The 
main challenge is to be meticulous about definitions, since some things 
are defined in terms of d/d(t) and some things are defined in terms of 
d/d(tau).

========

In any case, we have at least two (maybe three) different ways that
factors of gamma can crop up in the equations of motion, when the
equations are projected into the lab frame.  Trying to bury all that 
in the definition of "the" relativistic mass seems like a guaranteed 
losing proposition.

Life is soooo much simpler if we just use invariant mass, invariant
length, and invariant time.  That makes the physics simple.  At the 
end of the day, we can project the results into the lab frame if
necessary, which is just an exercise in geometry and trigonometry.