Einstein, in a 1905 paper on Special Relativity, introduced two seemingly innocent postulates:
one, that the velocity of light is the same in all inertial frames—and in particular
is the same in both directions. And by an inertial frame we mean any frame which is
at rest relative to the "fixed stars," or is moving at a uniform velocity with respect
to another inertial frame. And the other postulate is that the laws of physics are the same in
all inertial frames. Another way to say that is that it is impossible to tell by any experiment—including
experiments that use light—wether one is truly at rest or moving at a uniform velocity.
So 1) the velocity of light is the same in both directions and in all inertial frames,
and 2) it is impossible to tell wether you are truly at rest or moving at a uniform velocity.
Now, these seemingly innocent postulates have very unexpected consequences, and in an attempt
to derive these consequences Einstein often made use of "Gedankenexperiments," or thought
experiments, involving a moving train. And this apparatus is nothing more than a physical
realization of this abstract experiment that Einstein would do, which will help us to understand
some of the unexpected consequences of the two postulates of relativity. Now, what we'll
do is try to make measurements in two inertial frames, namely the "station frame" [laboratory
frame] and the "train frame," and try to relate the results of the measurements
in the two frames. In particular let's start off with trying to make this simple experiment
of measuring the length of a moving meter stick. Measuring the length of a fixed meter
stick—in other words one that is at rest in our own frame—is not a serious problem.
You just lay it down, pace off the distance between one end and the other, and that is
length of your fixed meter stick. The question of measuring the length of a moving meter
stick is a little bit more complicated, but we will choose the following scheme to measure the
length of a moving meter stick: we will say "Let's mark where its ends are at a particular
time, and then at our leisure pace-off the distance between these two marks, and claim
that to be the length of the moving meter stick." So let's do that experiment. Here
is a moving meter stick, let's mark where its ends are at a particular time, let's say
at quarter-of. Here it is quarter-of now and I put a marker, and at quarter-of here's a
clock over here, quarter-of I put a mark here, and now we can come and measure at our leisure
the distance between those two marks and say that that it the length of the moving meter
stick. Now some of you may have objected at this procedure, because you said "Well, we
didn't quite mark the ends at the same time." And I claim "Well, how can I tell time? I
used the two clocks, they both read the same time." And you say "A-ha! The clocks aren't
synchronized!" And you're right, they're not synchronized. They're just two random clocks,
they could be reading random times, just as they indeed are. So the question is how do
we synchronize two distant clocks? We have to be careful in synchronizing these two clocks
not to violate two postulates of relativity. And in particular we'll make use of the two
postulates of relativity to synchronize these two clocks. Let's assume that the two clocks
are an equal distance from the center of the station, and let's in fact place them at an
equal distance from the center of the station. There's that clock, and the right-hand clock
we'll place over here. Ok, so the two clocks are at the same distance from the center of
the station, and now if we set off a firecracker at the center of the station, and if the two
clocks are indeed synchronized, then they should receive the light at each end at exactly
the same time (the clocks should read the same time when they receive the flash of light
as it leaves from the center of the station) because we know the light travels with the
same speed in both directions, due to Einstein's first postulate. So let's
set off a firecracker in the middle of the station, and read at what time it arrives
here, and at what time it arrives at the other end. And the light arrives at this end...just
about noon, and it arrived here before noon. So obviously the two clocks were not synchronized,
because they didn't read the same time. And in fact this one arrived here about 10 minutes
too early, let me set it that way. And in fact you can see that these two clocks are
approximately synchronized. Now let me set them this way, and to do this accurately let's
in fact repeat the experiment. (Not so much to do it accurately but to do it properly—this
is not an experiment where we would get accurate results.) But anyway we set off a firecracker
in the middle, we step back and read what time each of these clocks read when the flash
of light reaches it. And this clock reads about 20-of and that one read about 20-of,
the two clocks are indeed synchronized. Ok, now this seems like such a trivial experiment
to do, you sort of wonder why we even bothered. After all, these two clocks appear to be going
together, you can see them ticking together. And in fact you can assume a whole ensemble
of clocks here, they would be ticking together and that's what you would normally mean when
you say the clocks are synchronized, they're all reading the same time. But the reason
why I was so careful in synchronizing these two clocks is that we will have to do the
same thing on the train. Remember Einstein's second postulate says it's impossible to tell
if you're truly at rest or moving, and it also says the [speed] of light is the same
in both directions no matter what inertial frame you're in. Let's do exactly the same
thing to see if clocks on the train are synchronized. We can set the clocks on the train moving.
Let's set off a firecracker at the middle of the train and see if the two clocks at
the beginning and at the end of the train are synchronized. Well the firecracker goes
off, this one reads about 20-of and this one...20-of is already passed and the flash still hasn't
arrived there, what conclusion do we draw? We draw exactly the same conclusion that we
did when the experiment was in the lab frame, that these two clocks are not synchronized.
They did not read the same time when the flash of light reached each end. And what we have
to do is we have to adjust one of the clocks and repeat the experiment until eventually
they read the same time. Let me set the time the way I believe it should be. And let me
also set the one in the middle, and let me prove later that the clocks are indeed synchronized.
Ok I claim that now these clocks are synchronized. Let's in fact verify this by repeating our
experiment. We'll set off a firecracker in the middle of the train and see what time
the flash reaches each end of the train. The firecracker went off, it reaches here at about
20-after, and the flash of light reaches the front end of the train...right about...now, which
is about 20-after. We therefore conclude that the clocks on the train, the front clock and
the rear clock, are now synchronized, because they read the same time as they should according
to Einstein's first postulate. The light travels the same speed in both directions. And we
know that the distance from here to the front and from here to the back are the same, therefore
they must read the same time. Now let's in fact verify that these two clocks are synchronized
with each other. We'll do exactly the same sort of experiment, we'll set off a firecracker at
the midpoint of those two clocks and see at what time the flash reaches each of these
two clocks. So the firecracker goes off, it reaches here at about quarter-after, and the
flash of light reaches here...at about quarter-after. So this clock and this clock are synchronized.
We'll leave it as an exercise to the student to prove that this clock and this clock are
also synchronized. So we now have a whole set of clocks that are synchronized in the
laboratory, and again picture in your mind's eye a whole ensemble of clocks all ticking
together. And it would be nice to picture an ensemble of clocks on the train—it's a
little more complicated to picture it, but nevertheless you can do so. But at least these
three clocks are synchronized with each other. And now let's go back to our experiment of
trying to measure the length of a moving meter stick. Ok, so we will now try to measure the
length of this moving meter stick in the lab frame by indicating where its ends are at
a particular instant of time, at a particular time according to our own clocks of course—the
laboratory clocks. So we will now try to measure the length of this moving meter stick in the
own clocks--the laboratory clocks. And we will do so by putting these markers down let's
say, to indicate where the ends of this meter stick are at midnight. This end is at here
at midnight, this end was of course here at midnight, because there was this imaginary
clock that read midnight when that point was over here. And now the distance between those
two markers is the length of that moving meter stick. Now the people on the train can do
the same thing. They can take this meter stick of ours and measure its length by marking
where in the train the ends of our meter stick are at a particular instant of time. So let's
assume they mark in their frame where its ends are at midnight. And the train starts
moving and here it is midnight and they put a marker over here and...midnight hasn't quite come
yet...here it is midnight right about here. That's this end was over here when this clock
read midnight. And now at their leisure they measure the distance between these two marks
with our meter stick and say that's the length of our meter stick. Now, notice something
very peculiar. One is that we measure the length of their meter stick to be approximately
the same as ours. The distance between these two marks is the length of their meter stick,
this is our meter stick, we measure the distance to be essentially the same. And they measure
the length of our meter stick to be the distance between these two marks, which is appreciably
shorter than their meter stick. Now, this is very serious, because it perhaps offers
some criterion of telling who's really at rest and who's moving! We could argue that
the people who measure the same length—namely us—are the ones who are truly at rest, and
the people who measure a shorter length—or the wrong length—are the ones that are moving,
thereby violating Einstein's second postulate of relativity, that it is possible to tell
if you're truly at rest or moving. But that's not where the flaw lies. The flaw is not in
Einstein's postulates but with these meter sticks. This was not the appropriate meter
stick for the people on the train—this was longer than a meter. If the people on the
train had used the appropriate meter stick—this one—then we would restore the symmetry. And
let me demonstrate that by re-doing the experiment. First we will measure the length of the train's
meter stick by indicating where its ends are in our frame at a particular instant of time
(at a particular time according to us). Ok let's start the train moving and mark where
its ends are at half-past. ... Let's mark where its ends are at quarter [of]. So at
quarter [of] this end was over here, and this end was over here, at quarter [of]. And we
say "All right, the length of the train's meter stick is the distance between these
two marks, and it is not the same as our meter stick—our meter stick is this long. The length
of the train's meter stick seems to be shorter than ours." Now let's repeat the experiment
and have the people on the train measure where the ends of our meter stick are at a particular instant
of time, let's say at half-past. At half-past this end was over here and at half past this
end is...over...here. So, they measure the length of our meter stick to be the distance
between those two yellow marks, and they compare it with their meter stick and conclude that
our meter stick is shorter than theirs. Now we have restored the symmetry! Namely, we
measure the length of their meter stick to be shorter than ours, they measure the length
of our meter stick to be shorter than theirs (and in fact if this were done properly the
ratio by which they are shorter would be exactly the same), and it is again impossible to tell
who's truly at rest and who's moving.