A Relativity Question

[Bowman_David/tiger_mpc@tiger.gtc.georgetown.ky.us]

Ed Schweber writes:  
> Hi fellow list members:
>
>   Here's a relativity question that's been bugging me for about
> a year now. I once posted it on the sci.physics Usnet group, but
> got only one response - which I frankly could not understand.
>
>   The speed of a transverse wave along a string is given by
>    
>             v = sqr root (T/u),
>
> where T is the tension and u is the linear mass density. Therefore,
> by chosing a large enough tension and a small enough linear mass density
> it should be possible to propagate a wave, and hence information,
> faster than the speed of light.
>
>   By first thought at a resolution was that this is making the mistake
> of applying a classical equation to a relativistic situation. But then I
> began to wonder if the classical approximation is necessarily invalid.
> After all, the speed of the wave propagation is not the speed at which
> any segment along the string would be moving. Any point along the string
> would have a speed on the order of the wave amplitude divided by the
> period, Since amplitude and period can be chosen independently of the wave
> speed it would seem that the speed at which any matter would be moving
> could be made arbitrarily small and the classical approximation as
> applicable as we want it to be.
>
>   Can anyone help me with the resolution. Thanks in advance.
>
> Ed Schweber (e-mail: EdSchweb@ix.netcom.com)
> Physics Teacher
> Solomon Schechter Day School
> West Orange, New Jersey

Ed, Your idea of reducing the linear mass density and increasing the tension
on the string will not work as the string will break at a wave speed many
orders of magnitude slower than c.  Let v = wave speed, e = the intermolecular
binding energy per bond, E_0 = the rest energy of each molecule.  It can be
shown with a straightforward order-of-magnitude physical argument that the
string will break when the tension on it is sufficient to give the ratio
v/c = order of(sqrt(e/E_0)).  The intermolecular binding energy per bond is
nowhere near the rest energy per particle and thus v << c.  Now you might
suppose that even though present-day materials have e << E_0 that maybe with
future technology we could make a material with e >= E_0.  This can't happen
because when e reaches E_0 the ground state energy is lowered (due to the
enormous binding energy) so much that the energy gap between the negative
energy hole (antiparticle) states and the positive energy particle states is
breached and the vacuum becomes unstable against pair creation.  (Look up the
Klein paradox regarding the Dirac equation.)  When this happens particle-
antiparticle pairs materialize between the original particles and the string
no longer holds together coherently.  A somewhat analogous phenomeon occurs in
hadrons when one tries to pull the constituent quarks apart in a baryon or
pull the antiquark away from the quark in a meson.  As these particles are
separated a quark-antiquark pair becomes energetically favorable over
continued increase in the potential energy tied up in the gluon field produced
by the separation of the color sources.  When this quark-antiquark pair
appears the original hadron breaks up into two hadrons, each of which has no
net "color".  (This is kind of *loosely* like trying to separate the North
pole of a bar magnet from the South pole by cutting the magnet in half.)

If anyone wants to see the argument as to why the string breaks when
v/c = order of(sqrt(e/E_0)) I can provide it in another post if there is
sufficient demand for it.

David Bowman
Georgetown College
dbowman@gtc.georgetown.ky.us