Re: [Phys-l] Re. Simultaneity

[John Denker <jsd@av8n.com>]

On 05/22/2009 06:32 AM, Bob Sciamanda wrote:
> Using w = ict, the usual Lorentz xform can be written as:
>
> x' = g(x+ivw/c),  and w' = g(w-ivx/c) where g =SQR(1/1-(v/c)^2)
>
> This can be written as the explicitly orthogonal xform:
>
> x' = xCOS(Q) + wSIN(Q) , and w'= wCOS(Q) - xSIN(Q),
>
> where Q is the imaginary angle :
> COS(Q) = g , and SIN(Q) = ivg/c

That can be written without complex numbers as

     x' = x cosh(ρ) + t sinh(ρ)
     t' = x sinh(ρ) + t cosh(ρ)

where ρ is the rapidity.  This expression is well-nigh unforgettable 
in analogy to the ordinary Euclidean rotation matrix.

Also 
  γ = cosh(ρ)
and
  β = v/c = tanh(ρ)     ("3-velocity")

All this is derived and discussed at
  http://www.av8n.com/physics/spacetime-trig.pdf


> one can maintain the orthogonality of x vs w and x' vs w' axis
> pairs in spacetime diagrams.

I am still skeptical of that statement about the "diagram".

We agree the math and the physics say that x' and t' are actually
orthogonal _in actual spacetime_.  So far so good.

OTOH what we draw on the diagram is not spacetime;  it is only the
_projection_ onto the plane of the diagram.  So what we draw is not 
orthogonal on the page (except in trivial cases).

You can't say a spacetime diagram "is" spacetime;  it is only an
imperfect representation of spacetime.  Orthogonal w.r.t the Minkowski 
metric in actual spacetime does not _look_ orthogonal w.r.t. the 
Euclidean metric on the page.

This is directly important because of the pedagogical and operational
issues pointed out by Michael E., having to do with finding coordinates
etc. either by projection onto "axes" or otherwise. (Preferably otherwise.)

   Note that none of the figures in 
     http://www.av8n.com/physics/spacetime-trig.pdf
   have axes ... only contours of constant x, constant t, et cetera.

=============

This general topic is as old as thought itself.  We are talking about 
the distinction between a _symbol_ and the _thing symbolized_.

 -- The expert looks at the symbol and thinks of the thing symbolized 
  (actual orthogonality in actual spacetime;  Minkowski metric). 
 -- Meanwhile the non-expert looks at the symbol and sees only the symbol 
  (concrete pixels on the page;  Euclidean metric on the page).


In a good model, some (most?) operations on the symbols directly represent 
operations on the things symbolized (e.g. following contours of constant x').

If the model is not perfect, there will be some operations on the symbols
that fail to represent operations on the things symbolized (e.g. projections
on the page that are orthogonal in the Euclidean representative space).

There are innumerable everyday real-world examples of this general topic.
For example, a car-key symbolizes ownership of the car.  But losing the
key does not mean forfeiting ownership of the car, finding a key does
not give you ownership of the car, and copying the key is much cheaper
than cloning the car.