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## Length contraction

So time is relative, what about distance? In order to think about this note that when we say that the distance between two objects is l we imagine measuring the position of these objects simultaneously...but simultaneity is relative, so we can expect distance to be a relative concept also.

To see this consider the above subatomic particles. As mentioned they are moving very fast but we can still imagine Superman (an unbiased observer if there is one) riding along with them. So we have two pictures: from the observer on earth Superman's clocks (accompanying the particle) are very slow, and so he/she can understand why it takes so long for the particle to decay. But for Superman the particle is at rest and so it must decay in its usual short time...the fact remains, however, that the particle does reach the earth. How can this be? Only if the distance which the particle traveled as measured in the frame of reference in which it is at rest is very short. This is the only way the observation that the particle reaches the earth's surface can be explained: for the observer on the earth this is because of time dilation, for the observer riding along with the particle, this is because of length contraction, see Fig. 6.13.

But we do not require peculiar subatomic particles in order to demonstrate length contraction (though the Principle of Relativity requires that if it occurs for the example above it should occur in all systems, otherwise we could determine by comparison which system has an absolute motion). So consider the previous experiment with the moving clock (Fig 6.12).

• The observer watching the clock move with velocity v notes that in a time T the clock moves a distance l=v T.
• The observer riding with the clock notes that the same distance is covered in a time T0; therefore the length measured by him/her is l0= v T0 (He also sees the other observer receding with speed v.)
• Therefore we have

Thus, the observer moving with the clock will measure a shorter length compared to the one measured by the other observer.

It is important to note that these expressions are not to be interpreted as illusions'', the an observer in motion with respect to a ruler will, when measuring its length, find a result smaller than the result of an observer at rest with respect to the ruler. An observer in motion with respect to a clock will measure a time larger than the ones measured by an observer at rest with the clock.

The question, what is `really' the length of a ruler?'' has no answer for this length depends on the relative velocity of the ruler to the measuring device . The same as with velocity, specifying lengths requires the framework provided by a frame of reference,

 Length is relative.

Note that this peculiar effect occurs only for lengths measured along the direction of motion and will not occur for lengths perpendicular to it. To see this imagine two identical trees, we sit at base of one and we observe the other move at constant speed with respect to us, its direction of motion is perpendicular to the trunk. In this setup as the roots of both trees coincide also will their tops, and so in both frames of reference we can simultaneously determine whether they have the same height; and they do. This is illustrated by the following 3 Quick-Time movie clips (you can download the player from Apple at http://www.apple.com/quicktime/ )

In the first the we are in a reference frame where both the C-shaped object and the box move, in the second we are on the box's reference frame, in the third on the C-shaped object's reference frame. In all cases the box fits nicely through the hole.

This implies that a moving object will be seen thinner (due to length contraction) but not shorter. Thin fellows will look positively gaunt at speeds close to that of light.

These conclusions require we also abandon Newton's description of space: distances are observer-dependent, no longer notches in absolute space.

Next: Paradoxes. Up: Enter Einstein Previous: The third prediction: The

Jose Wudka
9/24/1998