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Re: [Phys-l] how to explain relativity

On Jun 17, 2010, at 10:00 AM, John Denker wrote:

... in this
case there is no such thing as "the" rest frame of the
object.

Right.

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.

Right.

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.

Right (but just to be sure, I assume you are referring to the times of those two events as registered on clocks traveling with the two spaceships.)

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.

I can't make sense of this. I'm not 100% sure what you mean by congruent and I *really* don't understand what you mean by saying "the rope will lie along this contour." I THINK you are saying something like "consider the collection of pieces of rope constructed by requiring that the clocks traveling with each piece all read the same time."

Each molecule of the rope will have its own notion of proper
time,

Absolutely so.

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.

But here again you'll need to be clearer about what you mean by "congruent" for me to follow.

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.

Regardless of any possible confusion above, if your analysis suggests that an unstressed object whose ends begin accelerating simultaneously in its own original rest frame with identical proper accelerations directed along the line joining them doesn't develop internal tensile stress, then your analysis is mistaken. As explained in my previous message, references therein, and many more available via Google, the only way for an object to avoid developing internal stress is for it to undergo Born acceleration which requires progressively larger proper accelerations for more rearward portions of the object.

John Mallinckrodt
Cal Poly Pomona