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




Thanks very much for this John - it will take me a bit of time to digest and respond (if necessary), but I suspect it will clear most things up for me.

I would, however, like to respond to your issue about my use of the word 'absolute', as it is itself an interesting subject. You are pretty much correct in guessing why I used that adjective, and I agree it was a bad way of asking my question.

I'm simply hoping to express physical space-time concepts in the same style as Minkowski Space itself can be expressed - using definitions and axioms about abstract objects (called vectors) that have no internal structure. Minkowski Space takes its objects to be symbols A, B, C,... that form a vector space with a bilinear form satisfying this and that property. It's not just about mathematical elegance, though. I feel like as soon as you use components, you run the risk of blurring frame-dependent concepts and relationships with frame-independent ones. If you stick to component-free symbols like P=mU, you cannot possibly get confused and waste valuable time arguing about non-existent paradoxes.

Having said that, it just occurred to me that even in abstract Minkowski Space, the signature of the metric tensor gets defined in terms of vector components, and it does so without breaking ranks with the 'abstract' (earlier 'absolute') scheme, so my thinking was confused there.

Also, I wasn't looking for a photon's U here, by the way. Part of reaching an answer to a question is clarifying what it is you are really asking. I now think my issue is that we have a well defined vector P (for the photon) that cannot, it seems, be assigned to (or at least fruitfully pictured as residing in) the same vector space to which P (of the massive particle) belongs. Different (isomorphic copies of) vector spaces for different physical vectors maybe? Perhaps this is even wrong - I'll go away and think about your response, and my question, some more...

And thanks again for your thoughtful response. I am extremely pleased to have stumbled upon your writings and through them, this forum. There are even people who think critically and coherently about thermodynamics - the worst expressed physical theory of all IMHO!

I've been talking to myself unnecessarily for ten years it seems ;-)

Derek

Date: Mon, 27 Sep 2010 22:40:40 -0400
From: jsd@av8n.com
To: phys-l@carnot.physics.buffalo.edu
Subject: Re: [Phys-l] Absolute four-momentum of massless particles

On 09/27/2010 07:30 PM, Derek McKenzie wrote:

Like several on this list, I believe the best way to make relativity
perspicuous is to use its absolute-geometric form, rather than its
more popular relative-reference-frames formulation. However, it isn't
always easy to know how to represent frame-dependent concepts in a
coordinate free way, and this is one that I've been struggling with
of late.

Just to make sure we are all on the same page, let me declare (quite
uncontroversially, I hope) that a massive particle (i.e. one such
that m > 0), has a well-defined (time-like) world line in absolute
space-time (let's restrict ourselves to Minkowski Space here), and at
each event of its world-line we can define (intrinsically - using
differential geometry) a unit tangent vector U, from which we can
construct the vector P := mU and call it the (absolute) four-momentum
of the particle.

I must object to the word "absolute" above, and in the Subject:
line. I'm going to pretend it isn't there. There is no such
thing as "absolute space-time" or "absolute four-momentum".

Perhaps the intention was to encourage thinking about vectors as
abstract geometric and physical entities, rather than as lists
of components in this-or-that reference frame. The abstract
coordinate-free approach is elegant but not essential for asking
or answering the sort of question we have here.

But then we come to massless particles - the photon being our
prototype, of course. Now without rehashing the details, which have
already appeared on this list from time to time and are again bread
and butter stuff, the frame-dependent four momentum of such a
particle can be defined as (|p|,p), where p is the three-momentum of
the particle. But I now want to express this vector as an abstract
vector in Minkowski Space, just like I did with mU, and here's where
I run into a problem (hopefully, just reflecting my own confusion).
In particular, what is the equivalent object here to a 'unit tangent
vector on a particle's world-line'?

There isn't a unit tangent vector, and you don't need one.

The massive particle has, at each point, an inertial frame
comoving with it. The photon does not.

For a massive particle, the unit tangent vector represents its
proper time. For the photon, there is no such thing.

For the massive particle, p = m u. For the photon, there is
a perfectly good p and a perfectly good m=0, which should make
it obvious that p = m u is dead on arrival. Also u = dx/dτ and
the photon doesn't have any elapsed τ, so once again we see that
the idea of four-velocity is DoA.

For one thing, the vector has
zero-magnitude, which doesn't concern me in and of itself, except I'm
not sure then how to write it down abstractly. Also, such an object
doesn't seem to be unique. Wouldn't any null-vector along the
trajectory work (even if I could specify one, which I'm having
trouble doing).

Any vector in the general direction of the world line would
satisfy the condition that p•p = 0, but that is not the only
condition on p. We are also interested in p•x for various
vectors x. The /phase/ of the electromagnetic wave has
abstract geometric and physical significance, and the phase
goes like p•x/ℏ.

You may find the contours of constant phase (i.e. the wave
fronts) to be more informative than the 4-momentum vector.

The solution for a plane wave in spacetime is worked out in
some detail, including equations and including a spacetime
diagram of the wavefronts, at
http://www.av8n.com/physics/maxwell-ga.htm#sec-plane-waves

For the massive particle *and* for the photon, there is no
absolute p i.e. no invariant p. The most you can ask for is
for p to be covariant, so that p•x is invariant for any x of
interest.

The massive particle has a preferred frame where p=0, but
the photon does not.

Also it is tempting to think of something as "moving along"
its world line, but this is not acceptable for a null world
line. There is no elapsed time, so everything (including
phase) is constant along the world line.

Since none of the standard relativity tomes on my shelf (Rindler,
Misner et al, Wald, etc.) seem to describe the idea of an absolute
four-momentum vector for massless particles,

as indeed they should not

even though they do
discuss photons, I'm wondering if this issue points to something
deep. Could the problem be, for example, that there is no such thing
as a world-line of a massless particle?

There's a perfectly good world line. Just no 4-velocity and
no proper time.
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