This is a continuation of my previous note ... now focusing
on the conceptual and pedagogical issues.
This gets back to the first message in this thread: It really,
really pays to think of each relevant 4-vector as a first-class
geometrical and physical object unto itself. In particular, it
exists independently of whatever reference frame -- if any --
you choose.
In grade school, you learned to add vectors geometrically, tip
to tail, without reference to any reference frame or vector
basis.
This leads to a style of reasoning that helps keep you out of
trouble. That is, find a way of saying whatever needs to be
said using /manifestly/ invariant expressions. For example,
talking about the dot product of physically-relevant vectors
is manifestly invariant. In contrast, talking about the
components in this-or-that frame is not.
Sometimes it takes a bit of effort to find a component-free
manifestly-invariant expression. However, I find this to
be very cheap compared to (a) the cost of a mistake, or (b)
the cost of fussing with the component-based expression to
make sure there is no mistake.
This is why I like to emphasize that special relativity is
the geometry and trigonometry of spacetime -- nothing more,
nothing less. This leads to a /style of reasoning/ that
is both easier and more powerful than the component-based
approach that you see in typical introductory textbooks.
OTOH the component-based approach does have some advantages,
e.g. when grinding out numbers using a low-tech computer
program. (There exist computer-algebra systems that sorta
know about the vector-space axioms, but the ones I've seen
are not very user-friendly.)
On the third hand, once you have a geometry-based
expression, obtaining the corresponding component-based
expression is usually trivial. Just pick a basis and
turn the crank. For example, -p•u is a component-free
expression, but we can re-interpret it in component-based
terms: It is the energy in the rest frame. This is
super-easy, because the components of u are super-simple
in the rest frame.
There are standard tricks for converting things in the
other direction. For example, to create a component-free
expression for something that you understand in the frame
comoving with object A, you can build projection operators
based on the 4-velocity of A.
An expert should be able to use both representations,
and switch back and forth freely.
I say again, this leads to a /style of reasoning/. This style
has many advantages, including the fact that it exercises,
reinforces, and extends your intuition about vectors and basic
geometry.
Seriously: Rather than asking questions like "where is the mass"
it is better to ask "what are the 4-vectors doing". The 4-vectors
will tell you about the mass (and usually not vice versa).
Each of the relevant 4-vectors has a geometrical and physical
reality unto itself, independent of whatever reference frame
-- /if any/ -- you choose.
Sometimes students assume that visualizing what's going on
in four dimensions requires some supernatural ability. I
insist it is a skill that can be taught and can be learned.
It can't be learned overnight ... but over time you can
collect the tools and learn how to use them.