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Next: Space and Time Up: Enter Einstein Previous: Length contraction

Paradoxes.

The above conclusions can be very confusing so it might be worthwhile to discuss the a bit.

Take for example length contraction: the Principle of Relativity implies that if we measure the length some rod while at rest with respect to it, and then we measure it when it is moving along its length, the second measurement yields a smaller value. The crucial point to keep in mind is the condition that the first measurement is made at rest with respect to the rod.

Similarly suppose we have two clocks labeled 1 and 2. which are in perfect agreement when they are at rest with respect to each other. Suppose now these clocks are endowed with a relative velocity. Then when we look at clock 2 in the frame of reference in which clock 1 is at rest, clock 2 will be measured to tick slower compared to clock 1. Similarly, in the frame of reference in which clock 2 is stationary, clock 1 will run slower compared to clock 2.

These results can be traced back to the fact that simultaneous events are not preserved when we go from one reference frame to another.


There are many ``paradoxes'' which appear to imply that the Principle of Relativity is wrong. The do not, of course, but it is interesting to see how the Principle of Relativity defends itself.

1.
Consider a man running with a ladder of length l (measured at rest) and a barn also of length l (again, when measured at rest). The barn has two doors and there are two persons standing at each of them; the door nearer to the ladder is open the farthest is closed. Now the man with the ladder runs fast towards the barn while the door persons have agreed to close the first door and open the second door as soon as the rear of the ladder goes through the first door.
This is a paradox for the following reason. The ladder guy is in a frame of reference in which the ladder is at rest but the barn is moving toward him, hence he will find the length of the barn shortened (shorter than his ladder), and will conclude that the front of the ladder will hit the second door before the first door is closed.
The barn people in contrast find the ladder shortened and will conclude that it will fit comfortably. There will even be a short lapse between the closing of the first door and the opening of the second, there will be no crash and the ladder guy will sail through.
So who is right?
The answer can be found by remembering that an even simultaneous for the barn people (the closing and opening of the doors) will not be simultaneous for the ladder guy. So, while for the door person the opening of the rear door and closing of the front occur at the same time, the ladder guy will see the person at the second door open it before the person at his rear closes that door and so he will sail through but only because, he would argue, the door guards were not synchronized.


2.
There is an astronaut whose length is 6 ft and he sees a big slab of metal with which he/she is going to crash. This piece of metal has a square hole of length 6 ft. (measured at rest with respect to the slab). From the point of view of the astronaut the hole is shrunk and so he will be hit...and die! From the point of view of an observer on the shuttle the plate is falling toward earth and the astronaut moving at right angles toward it, hence this observer would measure a short astronaut (5 ft) [*] and conclude that he/she will not be harmed (see Fig. 6.14). What does really happen?

 
Figure 6.14: An astronaut's close encounter with a metal plate 
\begin{figure} \centerline{ \vbox to 3.8 truein{\epsfysize=4 truein\epsfbox[0 0 612 792]{6.str/astro.ps}} }\end{figure}


The problem is solved in the same way as above. For the astronaut to be hit a simultaneous coincidence of his head and legs with the two extremes of the slab's hole should occur. In fact he is not hit. What is more peculiar is what he sees: he will see the slab tilt in such a way that he goes through the hole with no problem!
This story illustrates the peculiar look which big objects acquire at very large speeds. For example, a kettle moving close to the speed of light with respect to, say, the Mad Hatter will be observed to twist in a very curious way indeed, see Fig. 6.15.

 
Figure 6.15: A relativistic kettle. The top view shows how the three dimensional view is distorted due to relativistic effects. The bottom view shows the corresponding behavior of a flat kettle which exhibits only length contraction.  
\begin{figure} \centerline{ \vbox to 4 truein{\epsfysize=5.6 truein\epsfbox[0 -100 612 692]{6.str/kettle.ps}} }\end{figure}


Just as for the case of length contraction and time dilation, the effect on the kettle is not an optical illusion, but any unbiased observer (such as a photographic camera) would detect the above images precisely as shown. If the relative velocity between the observer and the kettle is known, one can use the formulas of special relativity to determine the shape of the object when at rest with respect to it...and we would obtain the first of the images: a nice kettle


3.
Consider two identical twins. One goes to space on a round trip to Alpha-Centauri (the star nearest to the Sun) traveling at speeds very close to c. The round trip takes 10 years as clocked on Earth [*]. As seen by the twin remaining on Earth all clocks on the ship slow down, including the biological clocks. Therefore he expects his traveling twin to age less than 10 years (about 4.5 years for these speeds; the difference is large since the speed is close to c).
On the other hand the twin in the spacecraft sees his brother (a together with the rest of the solar system) traveling backwards also at speeds close to c and he argues that Einstein requires the twin on Earth to age less than 10 years. Thus each one states that the other will be younger when they meet again!
The solution lies in the fact that the traveling twin is not always in an inertial frame of reference: he must decelerate as he reaches Alpha-Centauri and then accelerate back. Because of this the expressions for time dilation as measured by the traveling twin will not coincide with the ones given above (which are true only for observers in different inertial frames). It is the traveling twin that will be younger.


next up previous contents
Next: Space and Time Up: Enter Einstein Previous: Length contraction

Jose Wudka
9/24/1998