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

[John Denker <jsd@av8n.com>]

On 09/30/2010 08:23 PM, Derek McKenzie wrote:
> Now this thread has got me wondering about a fact that I think is
> supposed to obvious, but isn't (to me). Massless particles must
> travel at the speed of light. Everybody assumes it, but what is/are
> the strongest argument/s for this assumption?
>
> One that comes to mind, given the arguments put forward in this
> thread, is that if a massless particle didn't travel at the speed of
> light, then its four momentum would have to be zero by virtue of the
> (now well-defined formula) P = mU, and conservation of four momentum
> would fail (I think). But this is a pretty weak argument because I
> can always declare that the formula P = mU only holds for massive
> particles (which is actually what we end up doing, in fact). There
> must be something really silly (i.e. inconsistent) about the very
> idea of a sub-luminal massless particle that I'm just not seeing.
> Which usually means I'm looking at things the wrong way...

You can prove that the massless particle (moving at
less than the speed of light) has zero energy and
zero momentum in all frames.  You can prove this in
about two lines, without assuming P = m U.

Hint:  What are the components of P in the rest frame
of the particle?

Take-home message:  There is a style of doing relativity
calculations that goes like this:  Calculate something 
in a frame where it is easy to calculate, then express 
it in a frame-independent way, whereupon it can be used 
in other frames.  A serious calculation may switch back
and forth many times, switching from frame-based to frame-
independent representations, using several different frames 
along the way, including the lab frame, the CM frame, et 
cetera.

Knowing how to express things in a frame-independent way 
is important, but it is not the only tool in the toolbox.