Relativistic Paradoxes Part C

From the sci.physics FAQ

The Superluminal Scissors

updated 31-MAR-1993

A Gedankenexperiment:
Imagine a huge pair of scissors, with blades one light-year long. The handle is only about two feet long, creating a huge lever arm, initially open by a few degrees. Then you suddenly close the scissors. This action takes about a tenth of a second. Doesn't the contact point where the two blades touch move down the blades much faster than the speed of light? After all, the scissors close in a tenth of a second, but the blades are a light-year long. That seems to mean that the contact point has moved down the blades at the remarkable speed of 10 light-years per second. This is more than 10^8 times the speed of light! But this seems to violate the most important rule of Special Relativity - no signal can travel faster than the speed of light. What's going on here?
We have mistakenly assumed that the scissors do in fact close when you close the handle. But, in fact, according to Special Relativity, this is not at all what happens. What does happen is that the blades of the scissors flex. No matter what material you use for the scissors, SR sets a theoretical upper limit to the rigidity of the material. In short, when you close the scissors, they bend.

The point at which the blades bend propagates down the blade at some speed less than the speed of light. On the near side of this point, the scissors are closed. On the far side of this point, the scissors remain open. You have, in fact, sent a kind of wave down the scissors, carrying the information that the scissors have been closed. But this wave does not travel faster than the speed of light. It will take at least one year for the tips of the blades, at the far end of the scissors, to feel any force whatsoever, and, ultimately, to come together to completely close the scissors.

As a practical matter, this theoretical upper limit to the rigidity of the metal in the scissors is far higher than the rigidity of any real material, so it would, in practice, take much much longer to close a real pair of metal scissors with blades as long as these.

One can analyze this problem microscopically as well. The electromagnetic force which binds the atoms of the scissors together propagates at the speeds of light. So if you displace some set of atoms in the scissor (such as the entire handles), the force will not propagate down the scissor instantaneously, This means that a scissor this big must cease to act as a rigid body. You can move parts of it without other parts moving at the same time. It takes some finite time for the changing forces on the scissor to propagate from atom to atom, letting the far tip of the blades "know" that the scissors have been closed.

The contact point where the two blades meet is not a physical object. So there is no fundamental reason why it could not move faster than the speed of light, provided that you arrange your experiment correctly. In fact it can be done with scissors provided that your scissors are short enough and wide open to start, very different conditions than those spelled out in the gedankenexperiment above. In this case it will take you quite a while to bring the blades together - more than enough time for light to travel to the tips of the scissors. When the blades finally come together, if they have the right shape, the contact point can indeed move faster than light.

Think about the simpler case of two rulers pinned together at an edge point at the ends. Slam the two rulers together and the contact point will move infinitely fast to the far end of the rulers at the instant they touch. So long as the rulers are short enough that contact does not happen until the signal propagates to the far ends of the rulers, the rulers will indeed be straight when they meet. Only if the rulers are too long will they be bent like our very long scissors, above, when they touch. The contact point can move faster than the speed of light, but the energy (or signal) of the closing force can not.

An analogy, equivalent in terms of information content, is, say, a line of strobe lights. You want to light them up one at a time, so that the `bright' spot travels faster than light. To do so, you can send a luminal signal down the line, telling each strobe light to wait a little while before flashing. If you decrease the wait time with each successive strobe light, the apparent bright spot will travel faster than light, since the strobes on the end didn't wait as long after getting the go-ahead, as did the ones at the beginning. But the bright spot can't pass the original signal, because then the strobe lights wouldn't know to flash.

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