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

[John Denker <jsd@av8n.com>]

I would like to emphasize that understanding arithmetic is a 
prerequisite for understanding relativity.

On Thu, September 30, 2010 1:33:59 AM I wrote in part:

> > Suppose the particle is moving with instantaneous 4-velocity v.
> > Let's calculate the invariant quantity P•v.  This is
> > easy, because P•v is just m times U•v and therefore
> > P•v is zero.

On 10/01/2010 11:15 AM, Moses Fayngold wrote:

> This is easy indeed, but wrong. You can write P*v = mU*v, but you
> cannot conclude from this that P*v = 0 even though m = 0. Such
> conclusion is based on unspoken assumption that U*v  IS FINITE. But
> precisely this assumption is wrong here. Even though the norm |U| =
> 1, its COMPONENTS are infinite for the photon (they are all
> proportional to the Lorentz factor - find its value for the speed of
> light!).

How many people on this list -- or their students -- are 
aware of any axioms of arithmetic that allow for multiplying
infinity by zero?  Not many, I would wager.

Just now I tried googling for such an axiom, and found nothing
... for good reason.

Let's do another little calculation.  Again we hypothesize that
  P = m U
  m = 0

We multiply both sides by 2.
  2 P = 2 m U

Since another of the axioms says that multiplication is associative, 
we can write this as
  2 P = (2 m) U

Now we also have
  2 m = m + m 
      = m + 0
      = m
where the last step follows from the axiom that says 0 is the additive 
identity.

Collecting results we have
  2 P = (2 m) U
      = m U
      = P

This in turn guarantees that P = 0.

So once again we have "proved" that every photon has zero energy and
zero momentum in every frame (subject to the hypothesis that P = m U
and m = 0).

Again we must reject the hypothesis.

==========

Maybe there could be some new set of axioms, unknown to mathematicians 
and unknown to physicists, such that 0 is not the additive identity.
This would allow us to write
  a) m = 0
  b) but 0 is not the additive identity.

The problem is, in the light of statement (b), statement (a) doesn't
mean anything.  I cannot imagine why anyone would want to go down
this road.

> This is one of the beautiful features of Relativity

I guess beauty is in the eye of the beholder.

==========

Bottom line:

Understanding arithmetic is a prerequisite for understanding relativity.