**Unformatted text preview: **3/19/2018 What time is it? previous home next Special Relativity: What Time is it?
Michael Fowler, University of Virginia
Table of Contents
Special Relativity in a Nutshell
A Simple but Reliable Clock
Looking at Somebody Else’s Clock
Fitzgerald Contraction
Experimental Evidence for Time Dilation: Dying Muons Special Relativity in a Nutshell
Einstein’s Theory of Special Relativity, discussed in the last lecture, may be summarized as follows:
The Laws of Physics are the same in any Inertial Frame of Reference.
(Such frames move at steady velocities with respect to each other.)
These Laws include in particular Maxwell’s Equations describing electric and magnetic fields, which
predict that light always travels at a particular speed c, equal to about 3×108 meters per second,
that is,186,300 miles per second.
It follows that any measurement of the speed of any flash of light by any observer in any inertial
frame will give the same answer c.
We have already noted one counter-intuitive consequence of this, that two different observers
moving relative to each other, each measuring the speed of the same blob of light relative to
him/herself, will both get c, even if their relative motion is in the same direction as the motion of the
blob of light.
We shall now explore how this simple assumption changes everything we thought we understood
about time and space. A Simple but Reliable Clock
We mentioned earlier that each of our (inertial) frames of reference is calibrated (had marks at
regular intervals along the walls) to measure distances, and has a clock to measure time. Let us
now get more specific about the clock—we want one that is easy to understand in any frame of
reference. Instead of a pendulum swinging back and forth, which wouldn’t work away from the
earth’s surface anyway, we have a blip of light bouncing back and forth between two mirrors facing
1/6 3/19/2018 What time is it? each other. We call this device a light clock. To really use it as a timing device we need some way
to count the bounces, so we position a photocell at the upper mirror, so that it catches the edge of
the blip of light. The photocell clicks when the light hits it, and this regular series of clicks drives the
clock hand around, just as for an ordinary clock. Of course, driving the photocell will eventually use
up the blip of light, so we also need some provision to reinforce the blip occasionally, such as a
strobe light set to flash just as it passes and thus add to the intensity of the light. Admittedly, this
may not be an easy way to build a clock, but the basic idea is simple. It’s easy to figure out how frequently our light clock clicks. If the two mirrors are a distance
w apart, the round trip distance for the blip from the photocell mirror to the other mirror and back is
2w. Since we know the blip always travels at c, we find the round trip time to be 2w/c, so this is
the time between clicks. This isn’t a very long time for a reasonable sized clock! The crystal in a
quartz watch “clicks “ of the order of 10,000 times a second. That would correspond to mirrors
about nine miles apart, so we need our clock to click about 1,000 times faster than that to get to a
reasonable size. Anyway, let us assume that such purely technical problems have been solved. Looking at Somebody Else’s Clock
Let us now consider two observers, Jack and Jill, each equipped with a calibrated inertial frame of
reference, and a light clock. To be specific, imagine Jack standing on the ground with his light clock
2/6 3/19/2018 What time is it? next to a straight railroad line, while Jill and her clock are on a large flatbed railroad wagon which is
moving down the track at a constant speed v : Jack now decides to check Jill’s light clock against
his own. He knows the time for his clock is 2w/c between clicks. Imagine it to be a slightly misty
day, so with binoculars he can actually see the blip of light bouncing between the mirrors of Jill’s
clock. How long does he think that blip takes to make a round trip? The one thing he’s sure of is
that it must be moving at c = 186, 300 miles per second, relative to him—that’s what Einstein tells
him. So to find the round trip time, all he needs is the round trip distance. This will not be
2w, because the mirrors are on the flatbed wagon moving down the track, so, relative to Jack on
the ground, when the blip gets back to the top mirror, that mirror has moved down the track some
since the blip left, so the blip actually follows a zigzag path as seen from the ground. Check out the animation!
Suppose now the blip in Jill’s clock on the moving flatbed wagon takes time t to get from the
bottom mirror to the top mirror as measured by Jack standing by the track. Then the length of the
“zig” from the bottom mirror to the top mirror is necessarily ct, since that is the distance covered by
any blip of light in time t. Meanwhile, the wagon has moved down the track a distance vt, where v
is the speed of the wagon. This should begin to look familiar—it is precisely the same as the
problem of the swimmer who swims at speed c relative to the water crossing a river flowing at v!
We have again a right-angled triangle with hypotenuse ct, and shorter sides vt and w.
3/6 3/19/2018 What time is it? From Pythagoras, then,
2 c t 2 2 = v t 2 2 + w , so
t 2 2 (c 2 2 − v ) = w , or
t 2 2 2 2 2 (1 − v /c ) = w /c , and, taking the square root of each side, then doubling to get the round trip time, we conclude that
Jack sees the time between clicks for Jill’s clock to be:
time between clicks for moving clock = 2w 1 c −
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√1 − v /c Of course, this gives the right answer 2w/c for a clock at rest, that is, v . = 0. This means that Jack sees Jill’s light clock to be going slow—a longer time between clicks—
compared to his own identical clock. Obviously, the effect is not dramatic at real railroad speeds.
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− The correction factor is √1 − v2 /c2 , which differs from 1 by about one part in a trillion even for a
bullet train! Nevertheless, the effect is real and can be measured, as we shall discuss later.
It is important to realize that the only reason we chose a light clock, as opposed to some other kind
of clock, is that its motion is very easy to analyze from a different frame. Jill could have a collection
of clocks on the wagon, and would synchronize them all. For example, she could hang her
wristwatch right next to the face of the light clock, and observe them together to be sure they
always showed the same time. Remember, in her frame her light clock clicks every
2w/c seconds, as it is designed to do. Observing this scene from his position beside the track,
Jack will see the synchronized light clock and wristwatch next to each other, and, of course, note
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− that the wristwatch is also running slow by the factor √1 − v2 /c2 . In fact, all her clocks, including
her pulse, are slowed down by this factor according to Jack. Jill is aging more slowly because
she’s moving!
But this isn’t the whole story—we must now turn everything around and look at it from Jill’s point of
view. Her inertial frame of reference is just as good as Jack’s. She sees his light clock to be
moving at speed v (backwards) so from her point of view his light blip takes the longer zigzag path,
which means his clock runs slower than hers. That is to say, each of them will see the other to
have slower clocks, and be aging more slowly. This phenomenon is called time dilation. It has
been verified in recent years by flying very accurate clocks around the world on jetliners and finding
they register less time, by the predicted amount, than identical clocks left on the ground. Time
4/6 3/19/2018 What time is it? dilation is also very easy to observe in elementary particle physics, as we shall discuss in the next
section. Fitzgerald Contraction
Consider now the following puzzle: suppose Jill’s clock is equipped with a device that stamps a
notch on the track once a second. How far apart are the notches? From Jill’s point of view, this is
pretty easy to answer. She sees the track passing under the wagon at v meters per second, so the
notches will of course be v meters apart. But Jack sees things differently. He sees Jill’s clocks to
be running slow, so he will see the notches to be stamped on the track at intervals of
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1/√1 − v /c seconds (so for a relativistic train going at v = 0.8c, the notches are stamped at
intervals of 5/3 = 1.67 seconds). Since Jack agrees with Jill that the relative speed of the wagon
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− and the track is v, he will assert the notches are not v meters apart, but v/√1 − v2 /c2 meters
apart, a greater distance. Who is right? It turns out that Jack is right, because the notches are in
his frame of reference, so he can wander over to them with a tape measure or whatever, and check
the distance. This implies that as a result of her motion, Jill observes the notches to be closer
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− together by a factor √1 − v2 /c2 than they would be at rest. This is called the Fitzgerald
contraction, and applies not just to the notches, but also to the track and to Jack—everything looks
somewhat squashed in the direction of motion! Experimental Evidence for Time Dilation: Dying Muons
The first clear example of time dilation was provided over fifty years ago by an experiment
detecting muons. These particles are produced at the outer edge of our atmosphere by incoming
cosmic rays hitting the first traces of air. They are unstable particles, with a “half-life” of 1.5
microseconds (1.5 millionths of a second), which means that if at a given time you have 100 of
them, 1.5 microseconds later you will have about 50, 1.5 microseconds after that 25, and so on.
Anyway, they are constantly being produced many miles up, and there is a constant rain of them
towards the surface of the earth, moving at very close to the speed of light. In 1941, a detector
placed near the top of Mount Washington (at 6000 feet above sea level) measured about 570
muons per hour coming in. Now these muons are raining down from above, but dying as they fall,
so if we move the detector to a lower altitude we expect it to detect fewer muons because a fraction
of those that came down past the 6000 foot level will die before they get to a lower altitude
detector. Approximating their speed by that of light, they are raining down at 186,300 miles per
second, which turns out to be, conveniently, about 1,000 feet per microsecond. Thus they should
reach the 4500 foot level 1.5 microseconds after passing the 6000 foot level, so, if half of them die
off in 1.5 microseconds, as claimed above, we should only expect to register about 570/2 = 285 per
hour with the same detector at this level. Dropping another 1500 feet, to the 3000 foot level, we
expect about 280/2 = 140 per hour, at 1500 feet about 70 per hour, and at ground level about 35
per hour. (We have rounded off some figures a bit, but this is reasonably close to the expected
value.)
5/6 3/19/2018 What time is it? To summarize: given the known rate at which these raining-down unstable muons decay, and given
that 570 per hour hit a detector near the top of Mount Washington, we only expect about 35 per
hour to survive down to sea level. In fact, when the detector was brought down to sea level, it
detected about 400 per hour! How did they survive? The reason they didn’t decay is that in their
frame of reference, much less time had passed. Their actual speed is about 0.994c, corresponding
to a time dilation factor of about 9, so in the 6 microsecond trip from the top of Mount Washington
to sea level, their clocks register only 6/9 = 0.67 microseconds. In this period of time, only about
one-quarter of them decay.
What does this look like from the muon’s point of view? How do they manage to get so far in so
little time? To them, Mount Washington and the earth’s surface are approaching at 0.994c, or
about 1,000 feet per microsecond. But in the 0.67 microseconds it takes them to get to sea level, it
would seem that to them sea level could only get 670 feet closer, so how could they travel the
whole 6000 feet from the top of Mount Washington? The answer is the Fitzgerald contraction. To
them, Mount Washington is squashed in a vertical direction (the direction of motion) by a factor of
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√1 − v /c , the same as the time dilation factor, which for the muons is about 9. So, to the
muons, Mount Washington is only 670 feet high—this is why they can get down it so fast!
Note: If you want to see how the experiment was actually carried out, with 1960 technology, in a
film made with 1960 technology, click here. Real insight into how cutting edge physics was done
back then...
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