Jean DAVID wrote:
I don't think that such string (inextensible) is physically impossible
because relavity laws must be observed.

Not because of relativity. Because an inextensible string is simply
physically impossible.

We can make shorter strings (or
rods) that can resist a certain tension of pulling.

Nope. The law governing the stretching or compressing (not breaking) of
strings, rods, cables, I-beams, columns, etc., is

(tension)/(cross-sectional area) = (material index)(change in
length)/(original length).

The only way (change in length) can be zero for a nonzero (tension) is
if the stiffness parameter (material index) is infinite. There is no
such beast, even theoretically. You'll also note that for a given
(tension), the *ratio* (change in length)/(original length) is a
constant. Thus, even if the (original length) is made short, you can
never make the (change in length) zero.

All of the above has nothing to do with relativity. *All* materials
stretch or compress a little, even if just a tiny amount of force is
applied.

Those objects don't have
to be perfectly rigid. I have supposed that the speed of unreeling of the
string is small enough so not to create a big tension to break the string.
The only difficulty I admit is the length of such string that we can product
for the experience.

You also said that speed of propagation in such a material is infinite, I
really doubt so.

Let's quell your doubts. Suppose you push in on one end of a metal bar,
and the question is how fast the other end of the bar moves in
response. How does the other end of the bar know that this end has been
pushed at all? The only way is if that information gets passed atom to
atom in the metal; that is, the first layer of atoms that gets pushed
then pushes the next layer of atoms, which pushes the next layer of
atoms, and so on. This propagation of the push to the other end of the
rod is *precisely* the speed of sound in the metal. So for the other
end to move instantaneously would require an infinite speed of
propagation. What happens in a *real* metal bar if you bang on one end?
The other end does not respond *at all* until the signal has propagated
there, and this takes a finite amount of time.

PD