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*From*: John Denker <jsd@av8n.com>*Date*: Thu, 20 Nov 2014 03:46:13 -0700

In the context of the "mixed representation" as set

forth at: http://www.sciamanda.com/rel.pdf

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.

**References**:**[Phys-L] Alternate Spacetime Diagram [was Trial balloon]***From:*Jeffrey Schnick <JSchnick@Anselm.Edu>

**Re: [Phys-L] Alternate Spacetime Diagram [was Trial balloon]***From:*"Bob Sciamanda" <treborsci@verizon.net>

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