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Re: [Phys-L] Alternate Spacetime Diagram [was Trial balloon]

In the context of the "mixed representation" as set
forth at:

First, here's where I'm coming from:
1) We're talking about a tool. Sometimes the choice of
tool comes down to a question of taste. De gustibus
non disputandum.

2) Having more tools is almost always a good thing in
principle. If a tool is useful to somebody at some
time, that's great, and the fact that it is less
useful in other situations is 100% irrelevant.

3) OTOH it is possible to compare one tool to another,
discussing the strengths and limitations of each.

4) As teachers we have a greater responsibility.
Sometimes we have to say "Personally I like this
tool, but you shouldn't get too enamored with it
because it comes with some terrible limitations."


In that spirit: The "mixed representation" comes with
some terrible costs and limitations.

In particular, I find that it obscures the fundamental
physics of the situation. Let me explain what I mean:

In grade school you presumably learned that a vector
is something with direction and magnitude. You can
add vectors geometrically, tip-to-tail, without
using anything resembling a frame of reference or
coordinate system. Note that Euclidean geometry
existed just fine for 2000 years before Fermat and
Descartes came along and introduced the idea of
coordinates. Furthermore, if/when you choose
to impose a frame of reference, the vector has
some interesting invariance properties: The
direction depends on your frame of reference, but
the magnitude does not.

By way of contrast, if you ask a C++ programmer
what a "vector" is, it's a list of objects, with
no connection to geometry of physics. Please,
let's not let this notion of "vector" degrade
our notion of real, geometrical, physical vectors.

In a conventional spacetime diagram, vectors have
nice geometrical properties. You can add vectors
tip-to-tail, without using anything resembling a
frame of reference or coordinate system. This is
a good thing!

The "trial balloon" says:

The particle world line (x = vt) is also the proper time axis of this particle, but
the quantitative use of this axis to indicate proper time (c) requires re-scaling by
the use of a cumbersome hyperbolic calibration curve.

I disagree. I say the particle (and its world line)
exist in physical terms, no matter what IF ANY coordinates
you choose to impose on the situation. I can impose the
blue coordinates, or the red coordinates, or both, or
neither, and the drawing of the particle (and its world
line) stays the same.

In contrast, to use mixed coordinates requires
redrawing the entire situation. This strikes me
as far more cumbersome than overlaying a suitably
scaled set of axes.

The idea that the events exist independent of any
coordinate system is simple, elegant, and tremendously
powerful. YMMV, but for me, it would take some really
special reason (e.g. a huge bribe) to get me to give up
this idea.

Also: If you find drawing the axes (or rather grids)
by hand to be cumbersome, get a computer to draw them
for you.

In this mixed space-time view the world line of a photon is
coincident with the x axis, illustrating that in the impossible photon frame, everything
“happens” at once.

What if I don't want it to be coincident with the x-axis?

The world-line of a photon is a 45° line in the red coordinate
system and also in the blue coordinate system. This to me is
the essence of relativity: Things look the same in any
coordinate system.

This trademark fact of special relativity cries out to be show-cased in any space-
time display as an obvious “Pythagorean” right triangle.

The way I draw spacetime diagrams, the Pythagorean
relationships are already showcased, no crying required.

If these relationships are not readily apparent, the
most likely cause is misreading the diagrams. It is
most definitely possible to misread a spacetime diagram,
especially if you use axes rather than grids.

Special relativity is the geometry and trigonometry of
spacetime ... nothing more and nothing less. The xt
plane uses hyperbolic trig functions in contrast to
the xy plane which uses the more familiar circular
trig functions, but overall, spacetime geometry is
remarkably similar to Euclidean geometry, way more
similar than it is different. The hyperbolas show
up super-clearly in an ordinary (non-mixed) diagram
and haven't seen anywhere near a strong enough
reason to give that up.


Last but not least, let me point out that spacetime
diagrams work just fine for [energy, momentum] 4-vectors,
not just [time, position] 4-vectors. Maybe I'm missing
something, but I don't see any way to get the mixed
representation to work for [E, p]. This seems like a
problem, because there is a rather solid connection
between position, velocity, and momentum.