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Re: A Relativity Question

On Fri, 12 Apr 1996, Edwin R. Schweber wrote:

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.

My 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...

I believe your first thought is correct. The wave equation for the string
with V = sqrt(T/u) (= wave velocity) is derived from a direct application
of the nonrelativistic version of Newton's 2nd law (mdv/dt = net force) to
each small bit of string mass, with the net force equal to the difference
in tension on the two sides of the bit of string and v = dy/dt the
vertical speed of each bit of string.

A quick relativistic calculation, putting the correct 4-velocity component
v/sqrt(1-v**2/c**2) of the bit of string into the second law and carrying
out the derivation, gives the wave speed as (T/u)*(sqrt(1-v**2/c**2))**3.
Since increasing the magnitude of string tension and/or diminishing the
mass density u of the string will tend to increase the vertical speed v of
each bit of string, the relativistic correction factor should keep the
wave speed under control.

A. R. Marlow E-MAIL: marlow@beta.loyno.edu
Department of Physics PHONE: (504) 865 3647 (Office)
Loyola University 865 2245 (Home)
New Orleans, LA 70118 FAX: (504) 865 2453