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*From*: "Paul Camp" <pjcamp@coastal.edu>*Date*: Sat, 13 Apr 1996 20:58:58 EST

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.

Ed Schweber (e-mail: EdSchweb@ix.netcom.com)

This is a _great_ question!

Leaving aside the matter of whether or not you could actually build

such a material, though, the traditional derivation of this

expression, based on Newton's 2nd Law, fails since the 2nd Law is not

a relativistically covariant expression (and in fact doesn't

generalize very well to 4 dimensions).

However, remember that what you have hold of here is the _phase_

velocity of the wave, omega/k, and it _is_ possible for that to exceed

the speed of light. Phase velocity measures the speed at which a

particular wave form propagates down the string and follows from

constancy of the phase angle, kx-(omega)t, for a specific point on the

wave form. This is what turns out to be determined by the tension

in the expression you gave above but to propagate at that velocity,

the wave can't change shape. This means you can't transport any

information at that speed.

Information is conveyed by changes in the wave form which propagate at

the _group_ velocity, d(omega)/dk, which agains follows from the

previous expression for the phase but this time allowing the values of

k and omega to vary (i.e. to think of the wave form as a superposition

of harmonic waves which reproduces the variation in shape you wish to

produce). The group velocity in fact must be bounded by c.

You cannot transport information by a harmonic wave -- it has no

beginning or end, goes on forever in all directions, and therefore

conveys no information other than the trivial fact that the wave maker

is turned on. To convey information, you must vary the wave somehow,

maybe switch it off and on, and then you no longer have a harmonic

wave with a definite omega and k but a superposition of many harmonic

waves, each with its own phase velocity. By switching it off and on,

however, you construct a wave packet and this packet propagates at the

group velocity. The fact that these two velocities are different, I

suppose, can be thought of as the reason that the wave packet will

eventually spread out rather than remaining tightly clumped together.

Furthermore, it turns out that energy is transported by a wave at the

group velocity and this gets us into relativity. While the force

concept does not generallize effectively to 4 dimensional spacetime,

the energy concept does and the relativistic energy expression tells

me directly that energy cannot propagate faster than c. So therefore

the group velocity is always less than c.

Paul J. Camp "The Beauty of the Universe

Assistant Professor of Physics consists not only of unity

Coastal Carolina University in variety but also of

Conway, SC 29526 variety in unity.

pjcamp@csd1.coastal.edu --Umberto Eco

(803)349-2227 The Name of the Rose

fax: (803)349-2926

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