Re: [Phys-L] The Physics Teacher and relativistic mass

[John Denker <jsd@av8n.com>]

On 07/27/2016 05:55 AM, Richard Tarara wrote:
> The 'real world' situation that has always seemed to me to call for
> velocity dependent mass is the following.  You have an electron (or
> any charged particle) accelerator and the ability to measure the
> velocity (in the lab frame) of the accelerated electrons.

It's ironic, because if you examine that system more carefully,
it is the poster child for why you do /not/ want to use
velocity-dependent mass.

  a) It takes some force in the dθ direction to bend the particle
   trajectory, to constrain it to follow the circular path.

  b) This is an /accelerator/ and it takes some force in the dr
   direction to accelerate the particle.

If we now define the velocity-dependent mass in the usual way, we
find *two different* values for it.  Back in the day, people tried
to define a «transverse» mass and a «longitudinal» mass.  This was
a huge mess, but it was the state of the art for about a year, more
than 100 years ago.  Then people got smart and started using the
invariant mass instead.

  https://www.av8n.com/physics/spacetime-welcome.htm
  https://www.av8n.com/physics/spacetime-acceleration.htm


On 07/27/2016 06:24 AM, Todd Pedlar wrote:

> > p is not equal to mv, but gamma * m * v.

That's true and entirely reasonable in context.

In a larger context I would add that it is even simpler to write:

	p = m u		[independent of whatever frame, if any, is used]
	u = dx/dτ	[ditto]

where τ is the proper time and u is the spacetime velocity, not
to be confused with the reduced velocity

	v = dx/dt	[in some particular chosen frame]

Rules of thumb:
 -- If you see a formula that has an explicit factor of γ, there is 
  usually a more-fundamental formula that doesn't.  Usually γ is
  the price you pay for using wall-clock time instead of proper time.
 -- In typical calculations, you can do all or most of the work without
  explicit γ factors.  If necessary, you can stick in a factor of γ
  (i.e. dt/dτ) at the very last step, if you want to project the
  results onto the laboratory frame.
 -- Bundling factors of γ into the mass is a terrible idea.