Star Trek Paradox

In the not too distant future, the starship, Enterprise, finds itself floating in space when its crew detects a Klingon Warbird approaching head-on at high velocity. Captain Kirk immediately raises the shields and prepares for battle. But before he can fire a shot, the Warbird zooms past, firing a photon torpedo at the instant of closest approach. Fortunately, the torpedo explodes harmlessly against the shield of the Enterprise, sending out a blinding burst of light in all directions that lasts for only a split second. Since the Warbird makes no attempt to maneuver to continue the battle, but continues on its course in a straight line at constant speed, Kirk decides not to retaliate by continuing the skirmish. However, he does protest the event in interplanetary court and brings the Klingon captain to trial.

In his defense, the Klingon maintains that his starship, not the Enterprise, was the victim; and that Kirk, not he, was at fault. He says that he was simply floating freely in space and the Enterprise attacked him head-on at high velocity. Yes, he fired a photon torpedo, but only in self-defense, and only at the very last possible instant in order to prevent his ship and crew from being destroyed by the Enterprise. He asserts, “It is Captain Kirk and his crew that should be on trial.”

But Kirk maintains that the Klingons were at fault. And to prove his point, he submits the following illustration to the court:

Fig. 1.1 The strafing incident from Captain Kirk’s point of view.

Fig. 1.1(a) illustrates the initial incident where the photon torpedo explodes against the shield of the Enterprise, sending out a burst of blinding light. Fig. 1.1(b) illustrates the situation at some arbitrary time later. From the second diagram it is clear that the Enterprise is at rest in space, because it remains forever centered on the three-dimensional spherical shell of expanding light (illustrated here as a two-dimensional circle). The Warbird, on the other hand, is clearly moving because it has moved off center while the circle of light expanded outward.

“Not so,” replies the Klingon captain. “Kirk’s drawings are all wrong. “ He then proceeds to submit the following illustration of the situation.

Fig. 1.2 The strafing incident from the Klingon captain’s point of view.

Although the first of these drawings is essentially the same as that of Captain Kirk (except for the arrow indicating who is moving), the second drawing is significantly different. According to the second drawing, the Warbird clearly is at rest in space, because it is the one that has not moved relative to the expanding circle of light. The Enterprise is moving, because it has moved off center while the circle of light expanded outward.

Now, the court is familiar with the Special Theory of Relativity and concludes that both of these diagrams are correct and that neither Kirk nor the Klingon captain is at fault. The whole incident was just an unfortunate accident that would best be forgotten, especially, since no one was hurt and no damage was done.

Analysis

Well, everything may be fine from the court’s point of view. And everything may be fine according to the Theory of Relativity. But I doubt very seriously if everything is fine with your understanding of the situation.

Tell me, if you will, how many centers does a circle (or shell) have? If Kirk’s diagram is correct, how can the Klingon’s diagram possibly be correct? How is it possible for both the Enterprise and the Warbird to remain at the center of the same circle of expanding light without staying together at the same point in space? (Remember, everyone agrees that the two spaceships really do separate after the initial incident.)

This simple thought experiment demonstrates clearly that a person can easily reach the correct answer to a relativistic problem without the slightest understanding about what really is going on. Without a global view of the situation, the Special Theory of Relativity presents apparent paradox after paradox. In fact, many students conclude that the whole Theory of Relativity is nothing more than an illusion. Since Relativity appears to bear no resemblance to common sense, students often find relativistic problems to be more confusing and difficult than they really should be. Consequently, many students tend to memorize the equations and techniques used to solve relativistic problems without ever developing an intuitive understanding of their basis. Don’t be one of these students. Learn both the equations and the concepts underlying those equations.

Questions:

Q1. The Star Trek paradox illustrates which of the following? (A) A circle, sphere, or shell has more than one center. (B) People can learn to give correct answers without understanding the situation at all. (C) Relativity contains irresolvable paradoxes and internal inconsistencies that render it philosophically an invalid description of nature. (D) More than one of these. (E) None of these.
B 514

ModPhy1/Unit1/Special Relativity/