Length Contraction

Length contraction (or to give it its formal name, the Lorentz-Fitzgerald contraction) is analogous to time dilation, in the sense that it is a phenomenon that is observed of objects in moving frames of reference. As in the previous section, consider an observer who is stationary in a particular frame of reference, which we will call frame A (e.g. someone standing still on a flat road - the road being the frame of reference). Suppose that this person has a rod that has length L. Now consider a second observer in a frame of reference (frame B) that is moving with respect to our original observer (e.g. in a car), and suppose that this observer also has a rod of length L, as measured in frame B.

The length contraction effect states that the observer in frame A observes the length of the rod in frame B as being shorter than the rod in frame A. The following diagram illustrates this.

It is very easy to show that the length of the rod in frame B, as measured by the observer in frame A, is related to the length of the same rod, as measured in frame B, through the equation

where v is the velocity with which frame B is moving, and C is the velocity of light. It is important to note that the length contraction only happens in the direction of motion of frame B. Thus if, as in the figures above, the motion is from left to right, then the length contraction occurs in the left-right direction, and not in the others (e.g. up-down).

Muons at the Earth's Surface

We will now consider a well-known experimental observation that requires us to invoke both time dilation and length contraction in order to explain the observation. But first, we require some background, to set the scene.

The earth is constantly being bombarded with particles and radiation from outer space known as "cosmic rays". Some of these cosmic rays interact with the upper atmosphere and generate other particles. The remainder pass directly down to the earth's surface, and indeed all humans receive a radiation dose from these cosmic rays. Some of the "secondary" particles created in the upper atmosphere decay in the upper atmosphere, and the remainder make it through the atmosphere and down to the surface of the earth.

Our interest lies in a type of particle called a "muon", which is one such particle generated in the upper atmosphere by incident cosmic rays. In a stationary frame of reference, muons have an average "life time" of 0.000002 seconds before they decay to something else. Muons generated by cosmic rays have a velocity of about 99.8% of the speed of light.

A simple piece of arithmetic shows us that such muons should travel a distance of

0.998 × 300,000,000 × 0.000002 = 600 metres

before they decay. Note that we have used here the fact that the speed of light, in SI units, is 300,000 kilometres per second = 300,000,000 metres per second.

This suggests that all cosmic muons should decay long before any of them can reach the surface of the earth, since the height of generation is an altitude of 6,000 metres or more. In theory, therefore, we should never observe cosmic muons at the earth's surface. However, large numbers of them are observed at the earth's surface, contrary to our argument above! The figure below illustrates this.

So what is the answer to this conundrum? In the section on time dilation, we showed that time runs more slowly in moving frames of reference, and this is the key to solving the conundrum. From our perspective on earth, the muon constitutes a moving frame of reference (even sub-atomic particles can be frames of reference), and what we are doing from earth is viewing a process that is occurring in a moving frame of reference, and which is therefore subject to the effects of time dilation.

To clarify this in the context of our discussion of time dilation, "frame A" is the surface of the earth, and "frame B" is the muon itself, which is moving relative to frame A at a speed of 99.8% of the speed of light.

We can use the formula for the magnitude of the time dilation effect,

with v = 0.998c and T_B = 0.000002 seconds and we find that to us, the life time of the speeding muon is not 0.000002 seconds, but 0.000032 seconds instead. That is, on account of its very high velocity relative to us, the lifetime of the muon appears to be 16 times greater. The range of a muon with this lifetime is given by

0.998 × 300,000,000 × 0.000032 = 9,600 metres

which is quite long enough for muons to reach the earth's surface, as we do indeed observe.

Resolution of a Paradox

So that solves the problem then. Well, not quite! You may (or may not) recall that in our discussion of the time dilation effect, we noted that the effect is reciprocal. That is, while our observer in frame A sees time running more slowly in frame B, the observer in frame B is equally entitled to say that frame A is moving with respect to him, and as such, the observer in frame B sees a clock in frame A running slowly compared with his own clock. The length contraction effect is reciprocal in the same way.

In the muon problem, the following issue arises. From the frame of reference of the muon, its half life is still 0.000002 seconds, and so as far as the muon is concerned (or if you prefer, from the perspective of someone "riding" on the back of the muon), it can only travel a distance of 600 metres from the point of generation, which as we have seen is not enough to reach the ground surface!

So what's the solution this time? The answer lies in the (reciprocal) length contraction of the distance between the point of muon generation and the earth's surface, as seen from the frame of reference of the muon. We can use the expression for length contraction above to show that the distance from point of creation to the earth's surface, as seen from the perspective of the muon, is equal to 600 metres, assuming that the muon has to travel 9,600 metres, in the frame of reference of the earth, in order to reach the surface of the earth.

So, even though the muon has a half-life on 0.000002 seconds in its own frame of reference, the distance it has to travel, from its frame of reference, is only 600 metres. It can accomplish this in 0.000002 seconds, at 99.8% of the speed of light.

So we see that the results of relativity, namely time dilation and length contraction, ensure that the appearance of cosmic muons at the earth's surface is quite reasonable, even though it would seem that their lifetime is far too short. The following diagram illustrates the situation, from the perspective of us on earth and from the perspective of the muon frame of reference.

Summary

A moving object appears to be shorter, in the direction of motion, compared with an identical object that is stationary with respect to the person making the observations. At normal speeds, the effect is very small, but as speeds approach those of the speed of light, the effect becomes more pronounced.

Next: Mass-Energy Relation