Re: [Phys-l] how to explain relativity

[John Denker <jsd@av8n.com>]

On 06/17/2010 09:17 AM, John Mallinckrodt wrote:

> > We can represent this
> > using a super-simple spacetime diagram:
> >
> >            A'         B'
> >           /          /
> >          /          /
> >          |          |
> >          |          |
> >          A          B
>
> Indeed, it is precisely this spacetime diagram that establishes the  
> constancy of the separation in the original rest frame.

So far so good.
 
> > The initial length of the rope is the proper distance between
> > A and B.  The final length of the rope is the proper distance
> > between A' and B'.  We can easily evaluate both of these lengths
> > in the lab frame.
>
> Yup.
>
> > But proper length is a Lorentz scalar, so
> > it is the same in /any/ frame.
>
> D'oh!  I was afraid this might be coming.  Much mischief results from  
> casual use of the idea of "proper length."  Better to stay away from  
> it.  It is the spacetime interval that is Lorentz invariant.  The  
> proper length of an object is determined by finding the interval  
> between two events that take place at the endpoints of an object *at  
> the same time* in the rest frame of the object.  In other frames, of  
> course, those events are not simultaneous.

That is not the definition of proper length, nor the only
way of measuring proper length.  In particular, in this 
case there is no such thing as "the" rest frame of the 
object.  We can -- with some difficulty -- restate this
crude idea into terms that make sense.

In all generality, the fact is that in spacetime, an event 
is an event, and the proper distance between two events is 
a Lorentz scalar.  This scalar can be evaluated in any 
convenient reference frame.  

For this problem, I find it convenient to use the lab frame.
Event A' is an event.  Event B' is an event.  The interval
between A' and B' is an invariant scalar.  This interval is
particularly easy to evaluate in the lab frame, since the
two events are simultaneous, i.e. they lie along a contour
of constant t.

They also (by construction) have the same proper time τ.

What's more, we can construct an entire contour of congruent
/local/ τ values, by interpolating many congruent copies of
the assigned proper acceleration profile a(τ).  If you do 
things correctly, the rope will lie along this contour.  
Each molecule of the rope will have its own notion of proper 
time, but if we make these congruent, as we should, then we 
can say -- in a loose but not fatally loose sense -- that 
we are measuring the proper length along a contour of constant 
proper time, keeping in mind that each molecule of the rope 
has its own notion of proper time, which we have carefully 
constructed in accordance with the "congruent motion" Ansatz.

Having done all this work, we find that the proper length of
the rope is |A'-B'| which is (by construction) the same as 
|A-B| ... as I have been saying all along.