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Applet of the experiment

# The Michelson-Morley Experiment

*Michael Fowler, U. Va. Physics*

## The Nature of Light

As a result of Michelson’s efforts in 1879, the speed of light was
known to be 186,350 miles per second with a likely error of around 30 miles per
second. This measurement, made by timing a flash of light travelling
between mirrors in Annapolis, agreed well with less direct measurements based
on astronomical observations. Still, this did not really clarify the *nature*
of light. Two hundred years earlier, Newton had suggested that light
consists of tiny* particles* generated in a hot object, which spray out at
very high speed, bounce off other objects, and are detected by our eyes. Newton’s arch-enemy Robert Hooke, on the other hand, thought that light must be a
kind of *wave motion*, like sound. To appreciate his point of view,
let us briefly review the nature of sound.

## Detecting the Aether Wind: the Michelson-Morley Experiment

Detecting the aether wind was the next challenge Michelson set himself after
his triumph in measuring the speed of light so accurately. Naturally,
something that allows solid bodies to pass through it freely is a little hard
to get a grip on. But Michelson realized that, just as the speed of sound
is relative to the air, so the speed of light must be relative to the aether. This
must mean, if you could measure the speed of light accurately enough, you could
measure the speed of light travelling upwind, and compare it with the speed of
light travelling downwind, and the difference of the two measurements should be
twice the windspeed. Unfortunately, it wasn’t that easy. All
the recent accurate measurements had used light travelling to a distant mirror
and coming back, so if there was an aether wind along the direction between the
mirrors, it would have opposite effects on the two parts of the measurement,
leaving a very small overall effect. There was no technically feasible
way to do a one-way determination of the speed of light.

At this point, Michelson had a very clever idea for detecting the aether
wind. As he explained to his children (according to his daughter), it was
based on the following puzzle:

*Suppose we have a river of width w (say, 100 feet), and two swimmers who
both swim at the same speed v feet per second (say, 5 feet per second). The
river is flowing at a steady rate, say 3 feet per second. The swimmers
race in the following way: they both start at the same point on one bank. One
swims directly across the river to the closest point on the opposite bank, then
turns around and swims back. The other stays on one side of the river,
swimming upstream a distance (measured along the bank) exactly equal to the
width of the river, then swims back to the start. Who wins?*

Let’s consider first the swimmer going upstream and back. Going
100 feet upstream, the speed relative to the bank is only 2 feet per second, so
that takes 50 seconds. Coming back, the speed is 8 feet per second, so it
takes 12.5 seconds, for a total time of 62.5 seconds.

The swimmer going across the flow is trickier. It won’t do
simply to aim directly for the opposite bank-the flow will carry the swimmer
downstream. To succeed in going directly across, the swimmer must
actually aim upstream at the correct angle (of course, a real swimmer would do
this automatically). Thus, the swimmer is going at 5 feet per second, at
an angle, relative to the river, and being carried downstream at a rate of 3
feet per second. If the angle is correctly chosen so that the net
movement is directly across, in one second the swimmer must have moved *four
feet* across: the distances covered in one second will form a 3,4,5
triangle. So, at a crossing rate of 4 feet per second, the swimmer gets
across in 25 seconds, and back in the same time, for a total time of 50
seconds. The cross-stream swimmer wins. This turns out to true
whatever their swimming speed. (Of course, the race is only possible if
they can swim faster than the current!)

Michelson’s great idea was to construct an exactly similar race for
pulses of light, with the aether wind playing the part of the river. The
scheme of the experiment is as follows: a pulse of light is directed at an
angle of 45 degrees at a half-silvered, half transparent mirror, so that half
the pulse goes on through the glass, half is reflected. These two
half-pulses are the two swimmers. They both go on to distant mirrors
which reflect them back to the half-silvered mirror. At this point, they
are again half reflected and half transmitted, but a telescope is placed behind
the half-silvered mirror as shown in the figure so that half of each half-pulse
will arrive in this telescope. Now, if there is an aether wind blowing,
someone looking through the telescope should see the halves of the two
half-pulses to arrive at slightly different times, since one would have gone
more upstream and back, one more across stream in general. To maximize
the effect, the whole apparatus, including the distant mirrors, was placed on a
large turntable so it could be swung around.

An animated applet of the experiment is available here–it
makes the account above a lot clearer!

Let us think about what kind of time delay we expect to find between the
arrival of the two half-pulses of light. Taking the speed of light to be *c*
miles per second relative to the aether, and the aether to be flowing at *v*
miles per second through the laboratory, to go a distance *w* miles
upstream will take *w*/(*c*-*v*) seconds, then to come back will
take *w*/(*c*+*v*) seconds. The total roundtrip time
upstream and downstream is the sum of these, which works out to be 2*wc*/(*c*²-*v*²),
which can also be written (2*w*/*c*)×1/(1-*v*²/*c*²).
Now, we can safely assume the speed of the aether is much less than the
speed of light, otherwise it would have been noticed long ago, for example in
timing of eclipses of Jupiter’s satellites. This means *v*²/*c*²
is a very small number, and we can use some handy mathematical facts to make
the algebra a bit easier. First, if *x* is very small compared to 1,
1/(1-*x*) is very close to 1+*x*. (You can check it with your
calculator.) Another fact we shall need in a minute is that for small *x*,
the square root of 1+*x* is very close to 1+*x*/2.

Putting all this together, the upstream--downstream roundtrip time

Now, what about the *cross-stream* time? The actual cross-stream speed
must be figured out as in the example above using a right-angled triangle, with
the hypoteneuse equal to the speed *c*, the shortest side the aether flow
speed *v*, and the other side the cross-stream speed we need to find the
time to get across. From Pythagoras’ theorem, then, the
cross-stream speed is the square root of (*c*²-*v*²).

Since this will be the same both ways, the roundtrip cross-stream time will
be

This can be written in the form

using the following approximations valid for low velocities,

Therefore the across-stream roundtrip time, assuming the aether velocity is much less than that of light, is

Looking at the two roundtrip times at the ends of the two paragraphs above,
we see that they differ by an amount (2*w*/*c*) × *v*²/2*c*².
Now, 2*w*/*c* is just the time the light would take if there
were no aether wind at all, say, a few millionths of a second. If we take
the aether windspeed to be equal to the earth’s speed in orbit, for
example, *v*/*c* is about 1/10,000, so *v*²/*c*²
is about 1/100,000,000. This means the time delay between the pulses
reflected from the different mirrors reaching the telescope is about
one-hundred-millionth of a few millionths of a second. It seems
completely hopeless that such a short time delay could be detected. However,
this turns out *not* to be the case, and Michelson was the first to figure
out how to do it. The trick is to use the *interference* properties
of the lightwaves. Instead of sending pulses of light, as we discussed
above, Michelson sent in a steady beam of light of a single color. This
can be visualized as a sequence of ingoing waves, with a wavelength one
fifty-thousandth of an inch or so. Now this sequence of waves is split
into two, and reflected as previously described. One set of waves goes
upstream and downstream, the other goes across stream and back. Finally,
they come together into the telescope and the eye. If the one that took
longer is half a wavelength behind, its troughs will be on top of the crests of
the first wave, they will cancel, and nothing will be seen. If the delay
is less than that, there will still be some dimming. However, slight
errors in the placement of the mirrors would have the same effect. This
is one reason why the apparatus is built to be rotated. On turning it
through 90 degrees, the upstream-downstream and the cross-stream waves change
places. Now the other one should be behind. Thus, if there is an
aether wind, if you watch through the telescope while you rotate the turntable,
you should expect to see variations in the brightness of the incoming light.

To magnify the time difference between the two paths, in the actual
experiment the light was reflected backwards and forwards several times, like a
several lap race.

Michelson calculated that an aether windspeed of only one or two miles a
second would have observable effects in this experiment, so if the aether
windspeed was comparable to the earth’s speed in orbit around the sun, it
would be easy to see. In fact, *nothing* was observed. The
light intensity did not vary at all. Some time later, the experiment was
redesigned so that an aether wind caused by the earth’s daily rotation
could be detected. Again, nothing was seen. Finally, Michelson
wondered if the aether was somehow getting stuck to the earth, like the air in
a below-decks cabin on a ship, so he redid the experiment on top of a high
mountain in California. Again, no aether wind was observed. It was
difficult to believe that the aether in the immediate vicinity of the earth was
stuck to it and moving with it, because light rays from stars would deflect as
they went from the moving faraway aether to the local stuck aether.

The only possible conclusion from this series of very difficult experiments
was that the whole concept of an all-pervading aether was wrong from the start.
Michelson was very reluctant to think along these lines. In fact,
new theoretical insight into the nature of light had arisen in the 1860’s
from the brilliant theoretical work of Maxwell, who had written down a set of
equations describing how electric and magnetic fields can give rise to each
other. He had discovered that his equations predicted there could be
waves made up of electric and magnetic fields, and the speed of these waves,
deduced from experiments on how these fields link together, would be 186,300
miles per second. This is, of course, the speed of light, so it is
natural to assume that light is made up of fast-varying electric and magnetic
fields. But this leads to a big problem: Maxwell’s equations
predict a definite speed for light, and it *is* the speed found by
measurements. But what is the speed to be measured relative to? The
whole point of bringing in the aether was to give a picture for light
resembling the one we understand for sound, compressional waves in a medium. The
speed of sound through air is measured relative to air. If the wind is
blowing towards you from the source of sound, you will hear the sound sooner. If
there isn’t an aether, though, this analogy doesn’t hold up. So
what does light travel at 186,300 miles per second relative to?

There is another obvious possibility, which is called the emitter theory: the
light travels at 186,300 miles per second relative to the source of the light. The
analogy here is between light emitted by a source and bullets emitted by a
machine gun. The bullets come out at a definite speed (called the muzzle
velocity) relative to the barrel of the gun. If the gun is mounted on the
front of a tank, which is moving forward, and the gun is pointing forward, then
relative to the ground the bullets are moving faster than they would if shot
from a tank at rest. The simplest way to test the emitter theory of
light, then, is to measure the speed of light emitted in the forward direction
by a flashlight moving in the forward direction, and see if it exceeds the
known speed of light by an amount equal to the speed of the flashlight. Actually,
this kind of direct test of the emitter theory only became experimentally
feasible in the nineteen-sixties. It is now possible to produce particles,
called neutral pions, which decay each one in a little explosion, emitting a
flash of light. It is also possible to have these pions moving forward at
185,000 miles per second when they self destruct, and to catch the light
emitted in the forward direction, and clock its speed. It is found that,
despite the expected boost from being emitted by a very fast source, the light
from the little explosions is going forward at the usual speed of 186,300 miles
per second. In the last century, the emitter theory was rejected because
it was thought the appearance of certain astronomical phenomena, such as double
stars, where two stars rotate around each other, would be affected. Those
arguments have since been criticized, but the pion test is unambiguous. The
definitive experiment was carried out by Alvager et al., *Physics Letters ***12**,
260 (1964).