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




John - I've been thinking through your post (below) on and off over the last couple of days and it has been a tremendous help, thank you!

Although I've got more work to do before I'm completely comfortable with the massless case, you've helped pinpoint a couple of conceptual trouble-makers. I'll mention them briefly here for the benefit of others who may have similar confusion, and also to confirm that my new thoughts are sensible...

1. When we learn differential geometry we often see the tangent vectors, and the ensuing tangent space, constructed through the notion of a parametrized curve. This is a great aid to intuition, particularly for physicists who like trajectories, but in massless particles we see a downside. The worldline can't be parametrized by proper time, because the latter is always zero (you could use a different parameter, but then you'd be doing mathematics rather than physics!). The tangent space still exists at each event of the particle's worldline - indeed it can be constructed in a very abstract way - so we still have access to the mountain of tensors we need to model physical properties, but these properties are pictured differently than in the massive case (if they can be pictured at all).

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, but by validating its existence at all we run the risk of incorrectly defining massless momentum as mU, which would then incorrectly equal zero. (Speaking of this result, I think some faulty arguments about this topic stem from people confusing a zero-length vector with the zero vector.)

3. By conferring a privileged status on the abstract definition of the four momentum of a massive particle, P = mU, over the componentized massless momentum vector, P = (|p|,p), I was deluded into thinking I was doing 'reference-frame-dependent stuff'. But using abstract components does not imply a reference-frame-dependent argument, as long as your argument does not tacitly assume a particular basis. I suspect this type of error might be very easy to commit.

4. Even though the constraint P.P = 0 does not pin down a photon's momentum vector singlehandedly (in the way P = mU does for massive particles), other physical constraints exist for determining P. In any case, the question of how to determine P should be kept separate from the argument for its existence.

5. I should stop using the term 'absolute' in these kinds of discussions. I get it from the philosophical literature which likes to distinguish 'absolute' and 'relational' spacetime theories, but in this context it can easily be confused with 'invariant'. (Actually, even in the sense in which I meant it I was incorrect).

Anyway, thanks again John - and more questions soon...

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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