Re: Bell's Spaceship paradox

On Feb 17, 4:54 am, jacques <jacques.f...@xxxxxxx> wrote:

I find this disymmetry
between distance and time intriguing and I wonder how physical is a
distance (therefore this stretch) in Relativity. The only physical
thing in relativity including SR looks to be "length" of worldline of
observers, the "s^2" as measured by clocks carried by the observers.
So I still wonder how physical is this stretch ?

Distances in SR (and hence stretching) are very physical. Give me two
points in space-time and I'll give you the distance between them.
Notice the careful wording. In SR, distances (or rather space-time
intervals) are defined between events (space-time points) rather than
"objects" (which occupy world-lines). Most of the confusion in puzzles
like Bell's spaceship paradox comes from forgetting this fact.

When two observers are stationary with respect to each other, it is
easy to decide how to define the distance between their world lines:
just take any space-like segment orthogonal to their world-lines (that
is, make use of their common notion of simultaneity). However, to
actually measure this distance, one has to come up with a local
experiment for either or both observers. One common trick is to
stretch a string between them. Assuming linear elasticity, the tension
in the string will be proportional to this distance. One must still
not forget that tension is a local quantity, which can potentially
vary with time and along the length of the string. Since the situation
is completely static, the tension is uniform both in time and along
its length (over the entire space-time world-*** swept out by the
string). So, a measurement of the tension by either observer, that is
at either end of the string, is enough to determine the distance
between them.

Unfortunately, once the observers are no longer mutually stationary,
the situation is no longer static. The string's tension becomes non-
uniform in time and along its length. So, the local tension measured
by either observer is no longer a reliable measure of any sort of
"instantaneous" distance between the observers. The moral of the story
is that, to correctly predict the outcome of a scenario such as Bell's
spaceship paradox, one must worry more about the tension suffered by
different parts of the string at various times rather than than what
the correct notion of simultaneity or distance should be (precisely
because these cannot be defined in a unique way between the two
observers).

So, here's an analysis of Bell's paradox from the above perspective.
The first step is to pick a model for the string, otherwise we cannot
say anything about its tension. The model does not need to be terribly
precise. For example, we can assume that the tension of a string
stretched between two mutually stationary observers is uniquely and
monotonically determined by the simultaneous distance between them
(which is perfectly well defined). Further, assume that there is a
critical tension, such that the string breaks as soon as this critical
tension is reached anywhere along its length (the breaking point will
be some space-time event). Finally, assume that disturbances propagate
along the string at some speed of sound, smaller than that of light,
and that their propagation is damped (so that we can recover the
previously described stationary behavior in case the ships eventually
become mutually stationary).

Consider the spaceship world-lines as shown in the Analysis section of
the Wikipedia article [1]. They consist of three segments: (a)
stationary in the lab frame, (b) accelerating, and (c) mutually
stationary but moving wrt the lab frame. In part (a), the string
tension is uniform and smaller than critical. In part (c), long after
the ships have stopped accelerating and the string vibrations have
died down, the string tension should again be uniquely determined by
the simultaneous distance between the ships' world-lines. So, if the
tension necessarily suffered by the string after a long enough time
exceeds critical, then the string must have broken at some prior
point, which would have been somewhere in the (b) or early stages of
the (c) sections.

To be any more precise about when and where the string breaks, one
would have to assume a specific dynamical model for the string and
solve its equations of motion with boundary conditions given by the
motion of the spaceships. For an extensive discussion of how this is
done (albeit in a more complicated context), see Greg Egan's thread
"Why is this model of relativistic elasticity flawed?" in the group
archives from last summer.

[1] http://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox

Hope this helps.

Igor

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