# Special Relativity: What Time is it?

*Michael Fowler, Physics Department, UVa. *

### 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 3×10^{8 }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 himself, 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 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 2*w*. Since we know the blip always travels at *c*, we find the round
trip time to be 2*w/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 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 2*w/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 2*w*, 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.

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*.

From Pythagoras, then,

*c*²*t*²
= *v*²*t*² + *w*²

so

*t*²(*c*²
- *v*²) = *w*²

or

*t*²(1 - *v*²/*c*²)
= *w*²/*c*²

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:

_{}

Of course, this gives the right answer 2*w*/*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. The
correction factor is _{}, 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 2*w/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 that the wristwatch is *also* running slow by the factor _{} 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
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 _{} seconds (so for a
relativistic train going at *v* = 0.8*c*, the notches are stamped at
intervals of 5/3 = 1.67 seconds). Since Jack agrees with Jill that the relative
speed of the wagon and the track is *v*, he will assert the notches are
not *v* meters apart, but _{} 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 together by a factor _{} 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

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 0.994*c*,
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,
_{} the same as the
time dilation factor, which for the muons is about 9. So, to the muons,