Pysics Insights

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 perfect: heavy stuff that slid or rolled frictionlessly, and connectors -- 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 must 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 sudden stops!