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Next: The third prediction: The Up: The second prediction: Simultaneity Previous: The first murder mystery

The second murder mystery (ca. 2330)

A murdered man is found in the cargo bay of the starship Enterprise with two head wounds caused by laser beams. The tragedy was observed from three places: a space station, the cargo bay itself, and a Klingon ship (a ``bird of prey''). At the time of the crime the Enterprise was moving at a speed c/2 with respect to the space station; the bird of prey was moving in the same direction as the Enterprise at a speed 3 c /4 with respect to the space station (and was ahead of the Enterprise). To simplify the language we will say that both ships as seen from the space station were moving to the right (see Fig. 6.6).


 
Figure 6.6: The setup for the second murder mystery. The velocities are measured with respect to the space station (labeled ``at rest'').  
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\centerline{ \vbox to 3.1 truein{\epsfysize=4 truein\epsfbox[0 0 612 792]{6.str/mm2_1.ps}} }\end{figure}

Everyone agrees that the dead man was hit on the head by two laser beams simultaneously. These beams were fired by a klingon at the back of the cargo bay, and by a human at the front. They shot while they stood at the same distance from the victim. Both life-forms are arrested and put to trial.

Captain Kirk, then at the space station, acts as the human's lawyer. Kirk points out that the klingon must have fired first. Indeed, at the time of the murder the klingon was placed in such a way that the Enterprise carried the victim away from his laser bolt; in contrast, the ship carried the victim towards the human's laser bolt (Fig. 6.7). Since both bolts hit at the same time, and they travel at the same speed c for all observers, the klingon must have fired first. ``The klingon's guilt is the greater one!'' Kirk shouted dramatically, and sat down.


 
Figure 6.7: Illustration of Kirk's argument (the murder as seen from the space station)  
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\centerline{ \vbox to 3.8 truein{\epsfysize=4 truein\epsfbox[0 0 612 792]{6.str/mm2_2.ps}}}\end{figure}

The captain of the bird-of-prey, who is (of course) acting as the klingon's lawyer, disagrees. His ship was moving to the right of the space station, but much faster than the Enterprise, hence, with respect to this ship, the Enterprise was moving to the left. ``I can then use my esteemed colleague's arguments and categorically state that it was the human that fired first (see Fig. 6.8), it is her guilt that is the greatest.''


 
Figure 6.8: Illustration of Klingon captain's argument (the murder as seen from the bird of prey)  
\begin{figure}
\centerline{ \vbox to 3.8 truein{\epsfysize=4 truein\epsfbox[0 0 612 792]{6.str/mm2_3.ps}} }\end{figure}


Dr. McCoy happened to be in the cargo bay at the time of the shooting and testifies that he saw both the human and the klingon fire at the same time: since the beams hit the victim at the same time, and they were at the same distance, they must have fired at the same time (Fig. 6.9).


 
Figure 6.9: Illustration of McCoy's argument (the murder as seen from the Enterprise).  
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\centerline{ \vbox to 3.8 truein{\epsfysize=4 truein\epsfbox[0 0 612 792]{6.str/mm2_4.ps}} }\end{figure}

Now, the law (in this story) states that the guilty party is the one who fired first, but deciding who did fire first is impossible! This is so because events occurring at different places will not be simultaneous to all observers. The fact that c is the same for all observers implies that if two events separated by some distance (such as the firing of the lasers) are simultaneous to one observer (such as McCoy) they will not be simultaneous to observers moving relative to the first (such as Kirk and the Klingon captain). Even the ordering in time of these events is relative

Simultaneity is relative for events separated by a non-zero distance. [*]

Let me use a short-hand and let K be the statement ``the klingon shoots'', while H denotes ``the human shoots''. Then


 
Figure 6.9: Illustration of McCoy's argument (the murder as seen from the Enterprise).  
Summary of the arguments        
K happens before H as seen from the space station (Kirk's argument)
H happens before K as seen from space station (Klingon capt.'s argument)
K simultaneous with H as seen from Enterprise (McCoy's argument)


So the Principle of Relativity forces us to conclude that in this situation the ordering of events in time is relative. But, this better not be true for all events: if the Principle of Relativity would predict that all time orderings are relative we could then imagine an observer who sees you, the reader, being born before your parents!

So there are events such as birth and death of a person which should occur in succession with the same ordering for any observer. And there are other events, such those in the shooting mystery, whose ordering in time is observer dependent. What is their difference?

The one clue is the following: in the story the assassins came in from opposite sides of a cargo bay and shot the victim. Since lasers travel at the speed of light, the human will receive the image of the klingon shooting only after she herself has fired (in order to see anything we must receive light from some source); the same is true for the klingon. So, when they fired they could not have been aware of each other's action.

This is not the same as for birth and death: a cat is born and then the dog eats the cat. It is then possible for you to tell your dog, that is, to send him a signal, that the cat was born. This signal reaches the dog before he performs his grim action (Fig. 6.10)


 
Figure 6.10: Illustration of events whose time ordering is the same for all observers.  
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\centerline{ \vbox to 3.9 truein{\epsfysize=4 truein\epsfbox[0 0 612 792]{6.str/sim.ps}} }\end{figure}

So two events A (cat is born) and B (dog eats cat) are ordered in the same way in time for all observers if we can send a signal at the time one event occurs (A) which will reach an observer who will witness the second event (B). In this case everyone will agree that A occurs before B, no matter what the relative speed of the observer. An extreme case consists of those events occurring at the same time at the same place will be seen to occur at the same time by all observers (everyone agrees that the laser beams hit the victim at the same time).

In contrast if no signals sent at the time A occurs can reach an observer before B happens, then the ordering in time of A and B depends on the relative velocity of the observer.

So there is no hope of going back in time with the winning Loto number and becoming a millionaire. If you think about it, the number of paradoxes which would arise if all time orderings were relative would be enormous: if you could go back in time, there would be two of you: one a pauper and the other a millionaire...but which one is you? Fortunately the Special Theory of Relativity simply disallows such situations.

Why did all this happen? Because the speed of light is always c. Both laser bolts will be seen to travel at the same speed by all observers, and because c is not infinite, the time it takes to reach a target depends on how the target is moving.

I will emphasize again the conclusions. Since the speed of light is the same for every observer in an inertial frame of reference, two things that are simultaneous to one observer will not be so according to other observers. The inescapable conclusion is that simultaneity is not an absolute concept: the statement ``two events at different places occurred at the same time'' is true only in a certain inertial reference frame and will be found to be incorrect in other frames.

Despite this there are events that everyone will agree are simultaneous: any two events happening at the same time and at the same spot will be seen to coincide by any observer. It is when the events are separated by a distance that simultaneity is relative. If events occurring at the same time and place for one observer were seen to occur at different times by another observer one can imagine going to a reference frame where the bullet that killed Lincoln went by his seat one hour before the president sat down. In this frame he was never assassinated!

One thing that Principle of Relativity does not permit is for some events which occur sequentially and such that the first affects the second to be inverted in order. For example it is impossible to go to a frame of reference in which the end of an exam occurs before it begins. It is only events that are mutually independent whose ordering in time can be inverted: two babies could be seen to be born one before the other or vice-versa, but only if they are not born at the same time at the same spot, so Jacob could not be the first born to Isaac (as opposed to Essau) in some frame of reference...the Bible's story is, in this sense, frame independent.


next up previous contents
Next: The third prediction: The Up: The second prediction: Simultaneity Previous: The first murder mystery
Jose Wudka
9/24/1998