Re: [Phys-l] Absolute four-momentum of massless particles

[John Denker <jsd@av8n.com>]

On 09/30/2010 08:30 AM, Derek McKenzie wrote:

> 2. It can be helpful to think of the four velocity vector as a
> 'proper time vector' so that we can see intuitively what goes wrong
> in the massless case (loosely - time stands still). I'm tempted to
> say the proper time vector of a photon is the zero vector,

Be careful there.  4-velocity is related to proper time,
but it's not the same thing.  Indeed it is _inversely_
related.  U goes like d/dτ, with τ in the denominator.

Let's see how that works, focusing on U rather than τ.
We consider the case of a particle with a very small but
nonzero mass.  I wish to _contrast_ the infinitesimal-mass 
particle with the zero-mass particle -- not equate them!

Suppose a highly relativistic particle whizzes past the 
laboratory.  We know the mass is very small but we don't
know how small.  We can measure the 4-momentum and we
find it is very close to P = [1, 1], well within the 
uncertainty of measurement.  If we are sure the mass is
nonzero we can write U = P/m = [1/m, 1/m] approximately,
subject to the constraint that U•U remains equal to -1,
which is no problem if we write U = [cosh ρ, sinh ρ] and
consider very large values of the rapidity ρ.  We see 
that the components of U are approaching infinity, not 
approaching zero.

For the small-mass particle, there will always be SOME
frame in which it is at rest.  For the zero-mass particle,
not so.  The limit as m goes to zero is a singular limit,
so intuition can be expected to fail, and you have to be
very careful about the limiting cases.

Saying a function has a singular limit means that the
value of f(0) is not equal to the limit of f(x) as x goes 
to zero.  For more on this, including the unforgettable
"worm in the apple" example, I highly recommend the 
article by Berry (yes, THAT Berry) at
  http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry341.pdf