How To Measure The Speed Of Light?

 The Speed of Light

Early Ideas about Light Propagation

As we shall soon see, attempts to measure the speed of light played an important part in the development of the theory of special relativity, and, indeed, the speed of light is central to the theory. 
The first recorded discussion of the speed of light (I think) is in Aristotle, where he quotes Empedocles as saying the light from the sun must take some time to reach the earth, but Aristotle himself apparently disagrees, and even Descartes thought that light traveled instantaneously.  Galileo, unfairly as usual, in Two New Sciences (page 42) has Simplicio stating the Aristotelian position,
SIMP.  Everyday experience shows that the propagation of light is instantaneous; for when we see a piece of artillery fired at great distance, the flash reaches our eyes without lapse of time; but the sound reaches the ear only after a noticeable interval. 
Of course, Galileo points out that in fact nothing about the speed of light can be deduced from this observation, except that light moves faster than sound.  He then goes on to suggest a possible way to measure the speed of light.  The idea is to have two people far away from each other, with covered lanterns.  One uncovers his lantern, then the other immediately uncovers his on seeing the light from the first.  This routine is to be practised with the two close together, so they will get used to the reaction times involved, then they are to do it two or three miles apart, or even further using telescopes, to see if the time interval is perceptibly lengthened.  Galileo claims he actually tried the experiment at distances less than a mile, and couldn’t detect a time lag.  From this one can certainly deduce that light travels at least ten times faster than sound. 

Measuring the Speed of Light with Jupiter’s Moons

The first real measurement of the speed of light came about half a century later, in 1676, by a Danish astronomer, Ole Römer, working at the Paris Observatory.  He had made a systematic study of Io, one of the moons of Jupiter, which was eclipsed by Jupiter at regular intervals, as Io went around Jupiter in a circular orbit at a steady rate.  Actually, Römer found, for several months the eclipses lagged more and more behind the expected time, but then they began to pick up again.  In September 1676,he correctly predicted that an eclipse on November 9 would be 10 minutes behind schedule.  This was indeed the case, to the surprise of his skeptical colleagues at the Royal Observatory in Paris.  Two weeks later, he told them what was happening: as the Earth and Jupiter moved in their orbits, the distance between them varied.  The light from Io (actually reflected sunlight, of course) took time to reach the earth, and took the longest time when the earth was furthest away.  When the Earth was furthest from Jupiter, there was an extra distance for light to travel equal to the diameter of the Earth’s orbit compared with the point of closest approach.  The observed eclipses were furthest behind the predicted times when the earth was furthest from Jupiter. 
From his observations, Römer concluded that light took about twenty-two minutes to cross the earth’s orbit.  This was something of an overestimate, and a few years later Newton wrote in the Principia (Book I, section XIV): “For it is now certain from the phenomena of Jupiter’s satellites, confirmed by the observations of different astronomers, that light is propagated in succession (note: I think this means at finite speed) and requires about seven or eight minutes to travel from the sun to the earth.”  This is essentially the correct value. 
Of course, to find the speed of light it was also necessary to know the distance from the earth to the sun.  During the 1670’s, attempts were made to measure the parallax of Mars, that is, how far it shifted against the background of distant stars when viewed simultaneously from two different places on earth at the same time.  This (very slight) shift could be used to find the distance of Mars from earth, and hence the distance to the sun, since all relative distances in the solar system had been established by observation and geometrical analysis.  According to Crowe (Modern Theories of the Universe, Dover, 1994, page 30), they concluded that the distance to the sun was between 40 and 90 million miles.  Measurements presumably converged on the correct value of about 93 million miles soon after that, because it appears Römer (or perhaps Huygens, using Römer’s data a short time later) used the correct value for the distance, since the speed of light was calculated to be 125,000 miles per second, about three-quarters of the correct value of 186,300 miles per second.  This error is fully accounted for by taking the time light needs to cross the earth’s orbit to be twenty-two minutes (as Römer did) instead of the correct value of sixteen minutes. 

Starlight and Rain

The next substantial improvement in measuring the speed of light took place in 1728, in England.  An astronomer James Bradley, sailing on the Thames with some friends, noticed that the little pennant on top of the mast changed position each time the boat put about, even though the wind was steady.  He thought of the boat as the earth in orbit, the wind as starlight coming from some distant star, and reasoned that the apparent direction the starlight was “blowing” in would depend on the way the earth was moving.  Another possible analogy is to imagine the starlight as a steady downpour of rain on a windless day, and to think of yourself as walking around a circular path at a steady pace.  The apparent direction of the incoming rain will not be vertically downwards—more will hit your front than your back.  In fact, if the rain is falling at, say, 15 mph, and you are walking at 3 mph, to you as observer the rain will be coming down at a slant so that it has a vertical speed of 15 mph, and a horizontal speed towards you of 3 mph.  Whether it is slanting down from the north or east or whatever at any given time depends on where you are on the circular path at that moment.  Bradley reasoned that the apparent direction of incoming starlight must vary in just this way, but the angular change would be a lot less dramatic.  The earth’s speed in orbit is about 18 miles per second, he knew from Römer’s work that light went at about 10,000 times that speed.  That meant that the angular variation in apparent incoming direction of starlight was about the magnitude of the small angle in a right-angled triangle with one side 10,000 times longer than the other, about one two-hundredth of a degree.  Notice this would have been just at the limits of Tycho’s measurements, but the advent of the telescope, and general improvements in engineering, meant this small angle was quite accurately measurable by Bradley’s time, and he found the velocity of light to be 185,000 miles per second, with an accuracy of about one percent. 

Fast Flickering Lanterns

The problem is, all these astronomical techniques do not have the appeal of Galileo’s idea of two guys with lanterns.  It would be reassuring to measure the speed of a beam of light between two points on the ground, rather than making somewhat indirect deductions based on apparent slight variations in the positions of stars.  We can see, though, that if the two lanterns are ten miles apart, the time lag is of order one-ten thousandth of a second, and it is difficult to see how to arrange that.  This technical problem was solved in France about 1850 by two rivals, Fizeau and Foucault, using slightly different techniques.  In Fizeau’s apparatus, a beam of light shone between the teeth of a rapidly rotating toothed wheel, so the “lantern” was constantly being covered and uncovered.  Instead of a second lantern far away, Fizeau simply had a mirror, reflecting the beam back, where it passed a second time between the teeth of the wheel.  The idea was, the blip of light that went out through one gap between teeth would only make it back through the same gap if the teeth had not had time to move over significantly during the round trip time to the far away mirror.  It was not difficult to make a wheel with a hundred teeth, and to rotate it hundreds of times a second, so the time for a tooth to move over could be arranged to be a fraction of one ten thousandth of a second.  The method worked.  Foucault’s method was based on the same general idea, but instead of a toothed wheel, he shone the beam on to a rotating mirror.  At one point in the mirror’s rotation, the reflected beam fell on a distant mirror, which reflected it right back to the rotating mirror, which meanwhile had turned through a small angle.  After this second reflection from the rotating mirror, the position of the beam was carefully measured.  This made it possible to figure out how far the mirror had turned during the time it took the light to make the round trip to the distant mirror, and since the rate of rotation of the mirror was known, the speed of light could be figured out.  These techniques gave the speed of light with an accuracy of about 1,000 miles per second. 

Albert Abraham Michelson

Albert Michelson was born in 1852 in Strzelno, Poland.  His father Samuel was a Jewish merchant, not a very safe thing to be at the time.  Purges of Jews were frequent in the neighboring towns and villages.  They decided to leave town.  Albert’s fourth birthday was celebrated in Murphy’s Camp, Calaveras County, about fifty miles south east of Sacramento, a place where five million dollars worth of gold dust was taken from one four acre lot.  Samuel prospered selling supplies to the miners.  When the gold ran out, the Michelsons moved to Virginia City, Nevada, on the Comstock lode, a silver mining town.  Albert went to high school in San Francisco.  In 1869, his father spotted an announcement in the local paper that Congressman Fitch would be appointing a candidate to the Naval Academy in Annapolis, and inviting applications.  Albert applied but did not get the appointment, which went instead to the son of a civil war veteran.  However, Albert knew that President Grant would also be appointing ten candidates himself, so he went east on the just opened continental railroad to try his luck.  Unknown to Michelson, Congressman Fitch wrote directly to Grant on his behalf, saying this would really help get the Nevada Jews into the Republican party.  This argument proved persuasive.  In fact, by the time Michelson met with Grant, all ten scholarships had been awarded, but the President somehow came up with another one.  Of the incoming class of ninety-two, four years later twenty-nine graduated.  Michelson placed first in optics, but twenty-fifth in seamanship.  The Superintendent of the Academy, Rear Admiral Worden, who had commanded the Monitor in its victory over the Merrimac, told Michelson: “If in the future you’d give less attention to those scientific things and more to your naval gunnery, there might come a time when you would know enough to be of some service to your country.”

Sailing the Silent Seas: Galilean Relativity

Shortly after graduation, Michelson was ordered aboard the USS Monongahela, a sailing ship, for a voyage through the Carribean and down to Rio.  According to the biography of Michelson written by his daughter (The Master of Light, by Dorothy Michelson Livingston, Chicago, 1973) he thought a lot as the ship glided across the quiet Caribbean about whether one could decide in a closed room inside the ship whether or not the vessel was moving.  In fact, his daughter quotes a famous passage from Galileo on just this point:
[SALV.] Shut yourself up with some friend in the largest room below decks of some large ship and there procure gnats, flies, and other such small winged creatures.  Also get a great tub full of water and within it put certain fishes; let also a certain bottle be hung up, which drop by drop lets forth its water into another narrow-necked bottle placed underneath.  Then, the ship lying still, observe how those small winged animals fly with like velocity towards all parts of the room; how the fish swim indifferently towards all sides; and how the distilling drops all fall into the bottle placed underneath.  And casting anything toward your friend, you need not throw it with more force one way than another, provided the distances be equal; and leaping with your legs together, you will reach as far one way as another.  Having observed all these particulars, though no man doubts that, so long as the vessel stands still, they ought to take place in this manner, make the ship move with what velocity you please, so long as the motion is uniform and not fluctuating this way and that.  You will not be able to discern the least alteration in all the forenamed effects, nor can you gather by any of them whether the ship moves or stands still.  ...in throwing something to your friend you do not need to throw harder if he is towards the front of the ship from you...  the drops from the upper bottle still fall into the lower bottle even though the ship may have moved many feet while the drop is in the air ...  Of this correspondence of effects the cause is that the ship’s motion is common to all the things contained in it and to the air also; I mean if those things be shut up in the room; but in case those things were above the deck in the open air, and not obliged to follow the course of the ship, differences would be observed, ...  smoke would stay behind...  . 
[SAGR.] Though it did not occur to me to try any of this out when I was at sea, I am sure you are right.  I remember being in my cabin wondering a hundred times whether the ship was moving or not, and sometimes I imagined it to be moving one way when in fact it was moving the other way.  I am therefore satisfied that no experiment that can be done in a closed cabin can determine the speed or direction of motion of a ship in steady motion. 
I have paraphrased this last remark somewhat to clarify it.  This conclusion of Galileo’s, that everything looks the same in a closed room moving at a steady speed as it does in a closed room at rest, is called The Principle of Galilean Relativity.  We shall be coming back to it. 

Michelson Measures the Speed of Light

On returning to Annapolis from the cruise, Michelson was commissioned Ensign, and in 1875 became an instructor in physics and chemistry at the Naval Academy, under Lieutenant Commander William Sampson.  Michelson met Mrs. Sampson’s niece, Margaret Heminway, daughter of a very successful Wall Street tycoon, who had built himself a granite castle in New Rochelle, NY.  Michelson married Margaret in an Episcopal service in New Rochelle in 1877. 
At work, lecture demonstrations had just been introduced at Annapolis.  Sampson suggested that it would be a good demonstration to measure the speed of light by Foucault’s method.  Michelson soon realized, on putting together the apparatus, that he could redesign it for much greater accuracy, but that would need money well beyond that available in the teaching demonstration budget.  He went and talked with his father in law, who agreed to put up $2,000.  Instead of Foucault’s 60 feet to the far mirror, Michelson had about 2,000 feet along the bank of the Severn, a distance he measured to one tenth of an inch.  He invested in very high quality lenses and mirrors to focus and reflect the beam.  His final result was 186,355 miles per second, with possible error of 30 miles per second or so.  This was twenty times more accurate than Foucault, made the New York Times, and Michelson was famous while still in his twenties.  In fact, this was accepted as the most accurate measurement of the speed of light for the next forty years, at which point Michelson measured it again.