Frictionless Elephants, Massless Strings, and Rigid Rods
"We're studying frictionless elephants and massless strings."
That's what some of us used to say about freshman physics, back when I
was in school. The point was that everything we studied was
heavy stuff that slid or rolled frictionlessly, and
-- massless strings and rigid rods -- whose physical properties were
ignored. The rods and strings were really just ways of describing
constraints on the equations.
In the real world, frictionless elephants wouldn't be able to stand up,
a ball of string certainly weighs something, and no rod is ever
perfectly rigid. None the less, these fictitious objects are
perfectly acceptable as far as Newtonian mechanics is concerned.
But not in relativity!
Oh, the elephants are still OK.
But the rigid rods and massless inextensible
strings which are
so popular in freshman mechanics have a fatal flaw. They carry
information from one end to the other in zero time. If you pull
on the end of an inextensible string, the other end immediately
transmits the pull to whatever it's tied to. If you push on one
end of a rigid rod, the other end immediately pushes on whatever it's
attached to. This is unphysical: Real pushes and pulls are
typically transmitted no faster than the speed of sound in the
material. But what's much worse, in relativity, instant
information transfer leads to contradictions. Information which
is transferred instantly from one place to another in an arbitrary
frame of reference can be viewed as traveling backwards in time in some
other frame of reference, and that's extremely
unphysical. (See The CCentipede
for one of my favorite examples.)
So, must we really abandon all use of ideal connectors, and specify
physical elasticity values for everything? This would surely make
describing any problem much harder!
The answer is "no". The situation isn't that dire. In fact,
there seems to be just one situation in which some care needs to be
exercised: sudden acceleration
. If velocities
are constant, then rigidity is, of course, irrelevant.
Furthermore, if acceleration is unvarying, then forces on the ends of
objects can have time to stabilize, and once again rigidity is
irrelevant. And if acceleration changes smoothly during the
problem, then we can also (usually) ignore the fact that the time to
transmit the infomation should have been longer than we assumed. But if
an acceleration is applied suddenly
to one part of a system,
and it's connected by inextensible members to other parts of the
system, then we may be in trouble, because the other parts of the
system will learn of the acceleration sooner than they should have.
The simplest example of this I know of is just the "guts" of the ccentipede
paradox. A "rigid rod" is
traveling, end-first, at relativistic velocity. It enters a
building through the front door. The building is too short to
hold the rod when both are stopped, but the contraction of the rod
makes it appear to fit in the building. Since the rod is
contracted, viewed from the building's frame of reference the whole
thing will fit at one time, so the valet who is standing by the door
waits until the rod is entirely inside the buildng, and closes the
door. Unfortunately, when the rod gets to the back wall of the
building, it hits the wall and stops, and that's a (very) sudden
acceleration. Because it's "rigid", the whole rod
stop at once -- which means, as viewed from the rod's frame, it stopped
before its back end ever entered the front door. This is a
contradiction, pure and simple. The problem is that we
transmitted information from the front of the rod to the back of the
rod faster than C, and in the frame of reference of the building, the
information actually moved backwards
in time (apply the Lorentz
transforms to it to verify that claim).
So once again, the point is not that rigid rods must never be used in
relativity problems. Rather, if you use them, watch out for those