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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.
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'?
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).
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,
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?