Notes on Relativity and Cosmology - Physics Department, UCSB

72 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
suggestion clearly fits with the Michelson-Morely experiment, we still have not
figured out how it fits with the stellar aberration experiments. So, we were
just exploring the suggestion to see where it leads. It led to a (ridiculous??)
prediction that clocks in different reference frames measure different amounts of
time to pass. Then, I tell you that this prediction has in fact been experimentally
tested, and that Einstein’s idea passed with flying colors. Now, you should begin
to believe that all of this crazy stuff really is true. Oh, and there will be plenty
more weird predictions and experimental verifications to come.
Another lovely example of this kind of thing comes from small subatomic particles
called muons (pronounced moo-ons). Muons are “unstable,” meaning that
they exist only for a short time and then turn into something else involving a
burst of radiation. You can think of them like little time bombs. They live (on
average) about 10 −6 seconds. Now, muons are created in the upper atmosphere
when a cosmic ray collides with the nucleus of some atom in the air (say, oxygen
or nitrogen). In the 1930’s, people noticed that these particles were traveling
down through the atmosphere and appearing in their physics labs. Now, the
atmosphere is about 30,000m tall, and these muons are created near the top.
The muons then travel downward at something close to the speed of light. Note
that, if they traveled at the speed of light 3 × 10 8 m/s, it would take them a
time t = 3 × 10 4 m/(3 × 10 8 m/s) = 10 −4 sec. to reach the earth. But, they are
only supposed to live for 10 −6 seconds! So, they should only make it 1/100 of
they way down before they explode. [By the way, the explosion times follow
an exponential distribution, so that the probability of a muon “getting lucky”
enough to last for 10 −4 seconds is e −100 ≈ 10 −30 . This is just about often
enough for you to expect it to happen a few times in the entire lifetime of the
Universe.]
The point is that the birth and death of a muon are like the ticks of its clock
and should be separated by 10 −6 seconds as measured in the rest frame of the
muon. In other words, the relevant concept here is 10 −6 seconds of proper time.
In our rest frame, we will measure a time 10 −6 sec/ √ 1 − (v/c) 2 to pass. For v
close enough to c, this can be as large (or larger than) 10 −4 seconds.
This concludes our first look at time dilation. In the section below, we turn
our attention to measurements of position and distance. However, there remain
several subtleties involving time dilation that we have not yet explored. We will
be revisiting the subject soon.
3.6 Length Contraction
In the last section, we learned how to relate times measured in different inertial
frames. Clearly, the next thing to understand is distance. While we had to work
fairly hard to compute the amount of time dilation that occurs, we will see that
the effect on distances follow quickly from our results for time.
Let’s suppose that two inertial observers both have measuring rods that are at
rest in their respective inertial frames. Each rod has length L in the frame in