Albert A. Michelson

How to use this course

Albert Michelson is remembered for an experiment that failed. In 1887 he and Edward Morley looked for the motion of the Earth through the luminiferous ether and found none, a result that relativity later made central. The rest of his career is just as remarkable. For fifty years he measured the speed of light more accurately than anyone else. He expressed the metre in wavelengths of light, measured the first stellar diameter, measured the rotation of the Earth with light, and built the instrument that now detects gravitational waves. His Nobel citation honoured his optical precision instruments and the measurements made with them.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Numbers are in SI units; where Michelson’s own figures are known, they are quoted and checked.

Timeline

From Strelno to the mining camps, 1852–1869

Albert Abraham Michelson was born on 19 December 1852 in Strelno, a small town in the Prussian province of Posen, now Strzelno in Poland. His family was Jewish and not religious; he remained an agnostic all his life. When he was about two the family emigrated to the United States, crossing the isthmus of Panama to reach California in the years of the gold rush. Records give the year as 1854 or 1855.

His father kept a dry-goods store in the mining town of Murphy’s Camp, in Calaveras County, and later moved the business to Virginia City, Nevada, the boom town of the Comstock silver lode. A local violinist taught the boy the violin, which he played for the rest of his life. His younger sister Miriam Michelson became a successful novelist and journalist.

There was no serious school in the camps, so Albert was sent to San Francisco for his secondary education. He lived for a time with the family of his high-school principal and earned a few dollars a month by looking after the school’s scientific apparatus. Millikan’s memoir of him records the arrangement, and it is the first sign of the experimenter he became.

Annapolis, 1869–1879

In 1869 Michelson competed for his Nevada district’s appointment to the U.S. Naval Academy and did not receive it. He travelled to Washington and obtained one of the at-large appointments in the gift of President Ulysses S. Grant. He entered the Academy as a cadet midshipman on 29 June 1869. He was strong in science and weak in seamanship, and he left the Academy in 1873 for two years at sea, which included a cruise in the West Indies. He was promoted ensign in 1874.

In December 1875 he returned to Annapolis as instructor in physics and chemistry. In April 1877 he married Margaret Hemingway, the daughter of a wealthy New York family; they had two sons and a daughter. Preparing a class demonstration of Foucault’s rotating-mirror method for the speed of light, he saw how to improve it enormously. He began trials in the winter of 1877–78 with apparatus costing a few dollars. In 1878 his father-in-law gave him $2000 for better equipment, and in June and July 1879 he made the measurements that made his name (Lesson 04). His result, published in 1880, was the most accurate yet obtained.

The same year he was posted to the Nautical Almanac Office in Washington, directed by Simon Newcomb, the leading American astronomer, who was planning his own measurement of the speed of light. Michelson worked with him on it before leaving for Europe.

Europe and Cleveland, 1880–1889

In 1880 Michelson took leave from the Navy to study in Europe, as ambitious American physicists then did. He spent the winter of 1880–81 in the Berlin laboratory of Hermann von Helmholtz, then studied at Heidelberg and in Paris at the Collège de France and the École polytechnique, where Alfred Cornu was among his teachers. In Berlin he designed an instrument to detect the Earth’s motion through the ether by splitting a beam of light into two perpendicular paths and recombining them. Street traffic made it unusable in Berlin, so in 1881 he moved it to the Astrophysical Observatory at Potsdam. He found no effect (Lesson 05). He resigned from the Navy in 1881.

In 1883 he became professor of physics at the new Case School of Applied Science in Cleveland, Ohio. Next door, at Western Reserve University, worked the chemist Edward Morley, a meticulous experimenter known for his determination of the atomic weight of oxygen. In 1884 Michelson repeated Foucault’s comparison of light in air and water, extending it to carbon disulphide. In 1886 he and Morley repeated Fizeau’s experiment on light in moving water with a much larger apparatus and confirmed Fresnel’s drag coefficient. Then, in the spring and summer of 1887, they performed the ether-drift experiment again with an instrument ten times more sensitive than the Potsdam one (Lesson 06).

The same year they proposed that a wavelength of light, rather than a metal bar, should serve as the ultimate standard of length. Michelson was elected to the National Academy of Sciences in 1888 and received the Rumford Prize of the American Academy of Arts and Sciences. In 1889 he accepted a professorship at the new Clark University in Worcester, Massachusetts, and moved there in 1890.

The rotating mirror, 1879

For the general reader. To time something as fast as light over a short distance, turn time into an angle. A mirror spinning hundreds of times a second reflects a beam to a distant mirror and back. By the time the light returns, the spinning mirror has turned slightly, so the returning beam comes back at a slightly different angle. Measure the angle and you know the travel time. Léon Foucault invented the method; Michelson made the deflection more than a hundred times larger.

The deflection

The factor 2 arises because a mirror turned through angle \(\alpha\) turns a reflected ray through \(2\alpha\), and the mirror turns through \(2\pi n\tau\) during the round trip.

Michelson’s improvement

In Foucault’s arrangement a lens formed an image of the source on a concave fixed mirror centred on the rotating mirror. The fixed mirror had to stay close, and Foucault folded his path with five concave mirrors to reach about 20 m. The image displacement was well under a millimetre. Michelson placed the rotating mirror near the focus of a lens of very long focal length (150 feet, about 46 m, made by Alvan Clark and Sons) and put a large plane mirror at the far end, perpendicular to the beam. The returning light was focused back near the slit whatever the distance, so \(D\) could be made large, and the displacement grew with both \(D\) and the radius \(r\). Michelson’s formula, in his own notation, is \(\tan\varphi=d/r\) with \(V=2592000''\,Dn/\varphi''\), where his \(D\) is twice the mirror separation; it is the same as the proposition.

The 1879 measurement

The mirrors stood 1986.23 feet (605.4 m) apart along the sea wall of the Naval Academy. The mirror turned at about 257 revolutions per second, timed against a tuning fork, and the radius was about 28 feet (8.6 m). The displacement of the slit image was over 100 mm, measured with a micrometer. After corrections for temperature and for the refractive index of air, his final value was

within 0.05% of the modern value. Newcomb’s more elaborate measurement in Washington (1880–82) gave 299 860 ± 30 km/s, and Michelson’s own repetition at Cleveland in 1882–83 gave 299 853 ± 60 km/s.

The interferometer, 1881

For the general reader. If light is a wave in an ether at rest, then the Earth, moving around the Sun at 30 km/s, moves through that ether, and light should take slightly longer to go out and back along the direction of motion than across it. The difference is one part in a hundred million, far beyond any clock. Michelson’s idea was to race two beams against each other and let interference compare their travel times to a fraction of a wavelength.

The instrument

Light from a source falls on a plate of glass half-silvered on one face, set at 45°. Half the light is transmitted to mirror \(M_1\), half reflected to mirror \(M_2\). Each beam returns along its path, and the plate recombines parts of both towards an observer. One beam passes through the glass plate three times and the other once, so Michelson added a compensating plate of the same glass in the second arm. Then the two arms differ only in their air paths, and white-light fringes can be found when the paths are equal.

Round trips in the ether

Suppose the apparatus moves through the ether at speed \(v\) parallel to arm 1, and write \(\beta=v/c\). Along arm 1 light travels at \(c-v\) going and \(c+v\) returning, relative to the apparatus:

For the transverse arm, in the ether frame the light must travel along the hypotenuse of a triangle while the mirror moves sideways, so its speed relative to the apparatus is \(\sqrt{c^2-v^2}\) each way:

In 1881 Michelson took the transverse time to be simply \(2L/c\), ignoring the sideways motion. That doubled the predicted effect. Alfred Potier pointed out the error in 1882 and Hendrik Lorentz analysed it in 1886. With arms of about 1.2 m, Michelson expected a shift of about 0.04 fringe; the corrected prediction was half that, comparable to the noise. His conclusion that the ether was carried along with the Earth was therefore not secure.

Why second order?

Experiments that compare a single light path with something else, such as Arago’s prism of 1810 or Fizeau’s and Hoek’s tests, are sensitive to first-order effects in \(\beta\). Fresnel’s partial drag (see the Fizeau course, Lesson 08) made all such first-order effects cancel. Only a round-trip experiment sensitive to \(\beta^2\approx10^{-8}\) could test the stationary ether, and James Clerk Maxwell had written in 1879 that such an effect was too small to measure. The interferometer was Michelson’s answer.

Michelson and Morley, 1887

For the general reader. In Cleveland in 1887 Michelson and Morley built an interferometer ten times more sensitive, floated it on mercury so that it could be turned smoothly, and watched the fringes as it rotated. The ether wind should have moved them by four tenths of a fringe. They moved by less than one hundredth. Within twenty years this null result had become one of the supports of Einstein’s special relativity.

The apparatus

The optics were mounted on a sandstone slab about 1.5 m square and 0.3 m thick, resting on a wooden float in a circular trough of mercury in the basement of a dormitory at Western Reserve University. Mirrors at the corners reflected each beam back and forth several times, giving an effective arm length \(L\approx11\) m. The slab turned slowly, about once in six minutes, and the observers read the fringe position at sixteen marked directions. Observations were made at noon and in the evening on several days in July 1887, so that the Earth’s orbital and rotational velocities would combine differently.

The contraction hypothesis

In 1889 George FitzGerald, and independently Lorentz in 1892, proposed that a body moving through the ether contracts along its motion by the factor \(\sqrt{1-\beta^2}\).

Proof. \(t_\parallel=\frac{2L\sqrt{1-\beta^2}}{c(1-\beta^2)}=\frac{2L}{c\sqrt{1-\beta^2}}=t_\perp\). The same holds for an arm at any angle to the motion (Exercise 6.3), so rotating the apparatus changes nothing.

Lorentz and others regarded the contraction as a physical effect of the ether on molecular forces. In 1905 Einstein derived it, together with time dilation, from two postulates: the principle of relativity and the constancy of the speed of light. In special relativity there is no ether wind to detect, and the round-trip speed of light in every direction is \(c\) in every inertial frame. Historians still debate how much the Michelson–Morley result itself influenced Einstein, who said at different times that it played little direct role in his thinking and that it had mattered. It was, in any case, the most cited experimental support for the theory, and Michelson himself remained attached to the ether.

Later repetitions

Morley and Dayton Miller repeated the experiment in 1902–04, again with a null result. Miller later claimed a small positive drift of about 10 km/s from observations on Mount Wilson in the 1920s; the result was never reproduced, and Georg Joos in 1930 set an upper limit of about 1.5 km/s. Roy Kennedy and Edward Thorndike in 1932 used unequal arms to test the combination of contraction and time dilation. Modern versions with optical resonators bound any anisotropy of the speed of light to parts in \(10^{17}\) or better.

Visibility, spectral lines and the metre

For the general reader. Fringes from a perfectly pure colour would persist however unequal the two arms of an interferometer. Real spectral lines have a width and often a hidden structure, so the fringes fade as the path difference grows. Michelson realized that the way they fade reveals the structure of the line. That let him find the purest line available, the red line of cadmium, and use it to measure the standard metre in wavelengths of light.

The visibility curve

Michelson defined the visibility of fringes as \(V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})\). Write the spectrum as a normalized intensity \(S(\sigma)\) in wavenumber \(\sigma=1/\lambda\), with \(\int S\,d\sigma=1\).

Proof. Each wavenumber contributes \(S(\sigma)\,(1+\cos2\pi\sigma x)\) and incoherent components add in intensity. Factor out \(e^{2\pi i\sigma_0x}\): the remaining integral \(\gamma(x)\) varies slowly with \(x\), while the fast oscillation \(\cos(2\pi\sigma_0x+\arg\gamma)\) swings between \(\pm1\). So \(I\) oscillates between \(I_0(1\pm|\gamma|)\), and \(V=|\gamma|\).

Michelson could measure only \(|\gamma|\), not its phase, and he inferred line profiles by guessing shapes that fitted the visibility curves, aided by his harmonic analyzer. With this method, in 1891–92, he showed that many lines thought simple were close doublets or had fine structure, among them the red line of hydrogen. The full method, recording \(I(x)\) and computing the spectrum by a Fourier transform, became Fourier-transform spectroscopy in the 1950s.

The metre in wavelengths

At the invitation of the International Committee for Weights and Measures, Michelson worked at the International Bureau at Sèvres in 1892–93 with Jean-René Benoît. Nobody could count a million and a half fringes directly, so Michelson built a chain of nine intermediate standards, or étalons, each about twice the length of the previous, from a fraction of a millimetre up to 10 cm. The fringes were counted for the shortest étalon; each longer one was compared with twice the previous by interference, with only a small fractional excess to measure; the 10 cm étalon was stepped ten times along the metre. They used the red, green and blue lines of cadmium, chosen because their visibility curves showed them to be nearly simple. The red line came out at about 643.85 nm in standard air, about 1 553 164 wavelengths to the metre. In 1907 the cadmium red line was adopted to define the ångström, the spectroscopists’ unit, and in 1960 the metre itself was redefined by a wavelength of krypton-86, compared through the chain back to cadmium.

The echelon

In 1898 Michelson introduced the echelon spectroscope: a stack of \(N\) glass plates of equal thickness \(t\), each set back from the next by a fixed step, forming a staircase. Light through successive steps acquires path differences of about \((n-1)t\), so the echelon is a grating with few lines working in very high order.

As for any grating, the resolving power is the order times the number of elements; the order is the path difference per step in wavelengths, and dispersion of the glass adds the second term. Michelson’s later work on ruling engines produced large diffraction gratings: after about eight years of work, a 6-inch grating with 110 000 lines, and 8-inch and 10-inch gratings by 1915.

Measuring the stars

For the general reader. Fizeau had suggested in 1868 that interference could measure the sizes of stars. Michelson worked out the theory independently in 1890 and showed how to do it: watch the fringes from two separated mirrors and increase their separation until the fringes vanish. He measured the moons of Jupiter in 1891. In 1920, on the 100-inch telescope at Mount Wilson, he and Francis Pease measured the giant star Betelgeuse, the first star other than the Sun whose size was known.

The visibility of fringes from two apertures a distance \(B\) apart is the Fourier transform of the source brightness at spatial frequency \(B/\lambda\) (the van Cittert–Zernike theorem, stated in Lesson 10 of the Fizeau course). This is the angular analogue of Lesson 07, where the visibility was the Fourier transform of the spectrum.

Derivation. The disc has brightness \(1\) for \(|\boldsymbol\alpha|\le\theta/2\). Its two-dimensional Fourier transform at spatial frequency \(q=B/\lambda\) is \(\int_0^{\theta/2}\!\int_0^{2\pi}e^{-2\pi iq\rho\cos\phi}\rho\,d\phi\,d\rho=2\pi\int_0^{\theta/2}J_0(2\pi q\rho)\rho\,d\rho\). Using \(\int_0^a J_0(k\rho)\rho\,d\rho=aJ_1(ka)/k\) and normalizing by the area \(\pi\theta^2/4\) gives \(2J_1(x)/x\) with \(x=\pi q\theta\). The first zero of \(J_1\) is at \(x=3.832\), so \(B=3.832\lambda/(\pi\theta)=1.22\lambda/\theta\).

Jupiter’s satellites and Betelgeuse

In 1891, at Lick Observatory, Michelson placed a mask with two adjustable slits over the 12-inch refractor and measured the diameters of the four Galilean satellites, in good agreement with micrometer measurements. For stars a much longer baseline was needed. In 1920 a steel beam 20 feet (about 6 m) long was mounted across the top of the 100-inch Hooker telescope, carrying two movable outer mirrors that fed light to two inner mirrors and so into the telescope. The telescope only had to bring the two beams together; the baseline was set by the outer mirrors. Pease made most of the observations. On 13 December 1920 the fringes of Betelgeuse vanished at a separation of about 121 inches (3.07 m), giving an angular diameter of 0.047 arcseconds. With the distance then estimated, the star was about 240 million miles across, larger than the orbit of the Earth.

The method works through turbulence because the atmosphere shifts the fringes but, for small mirrors, hardly reduces their contrast. Larger beams proved mechanically unstable, and stellar interferometry stalled until the 1970s, when separate telescopes were combined with laser-controlled delay lines. Modern arrays such as the CHARA array on Mount Wilson and the VLT Interferometer are its direct descendants.

The rotating and the yielding Earth

For the general reader. Two of Michelson’s late experiments used light to study the Earth itself. In the first, with Henry Gale, long pipes half filled with water served as giant levels, and interference measured how the Moon and Sun tilt the ground. The result showed that the Earth is as rigid as steel. In the second, light sent in both directions around a rectangle of evacuated pipe more than a kilometre and a half around showed the rotation of the Earth, without looking at the sky.

The Sagnac effect

Derivation for a circle. Let the loop be a circle of radius \(R\), rotating with rim speed \(v=\Omega R\), and work in the inertial frame. The co-rotating beam must cover \(2\pi R+v t_+\), so \(t_+=2\pi R/(c-v)\); the counter-rotating beam needs \(t_-=2\pi R/(c+v)\). Then \(\Delta t=\frac{4\pi Rv}{c^2-v^2}\approx\frac{4\pi R^2\Omega}{c^2}=\frac{4A\Omega}{c^2}\). The result holds for any shape by applying Stokes’s theorem to \(\oint\mathbf v\cdot d\mathbf l=2\Omega A\).

Michelson proposed the experiment in 1904. It was carried out in 1925 at Clearing, Illinois (latitude 41°46′), with Gale and Fred Pearson. A rectangle of 12-inch water pipe, about 613 m by 339 m, was evacuated, and a much smaller rectangle sharing one side served as a reference, since the large loop could not be rotated to set a zero. The predicted shift was 0.237 fringe; the observed shift was 0.230 ± 0.005. Ring-laser gyroscopes and fibre-optic gyroscopes in aircraft use the same effect.

Earth tides

From 1913 Michelson and Gale buried two long iron pipes, one running north–south and one east–west, each half filled with water. As the Moon and Sun pulled on the Earth, the water surface tilted relative to the pipe. Interferometers at the ends measured the water level to small fractions of a wavelength. The observed tides were somewhat smaller than those expected for a perfectly rigid Earth, and the ratio measured the Earth’s elastic yielding. The results, published in 1919, implied that the Earth as a whole is about as rigid as steel.

Mount Wilson and the last measurements of c

For the general reader. In his seventies Michelson returned to the speed of light. He sent a beam from Mount Wilson to Mount San Antonio, 35 km away, and back, using a spinning many-sided mirror. The distance was surveyed by the Coast and Geodetic Survey to a few parts in a million. His 1926 result was within a few kilometres per second of today’s value. His final experiment, in a mile-long evacuated pipe, was finished after his death.

Instead of measuring a small angle, Michelson adjusted the speed until the image was stationary, timing the rotation precisely, and measured only the small residual displacement. The precision then rested on the rotation rate and the distance. With \(m=8\) and \(D\approx35.4\) km the null speed is about 529 revolutions per second. He used prisms of glass and steel with 8, 12 and 16 faces, and the agreement among them tested systematic errors. The 1926 series gave

A practical difficulty was the refractive index of air over a 35 km mountain path, which had to be estimated from temperatures and pressures at the two ends. To remove it, in 1929 Michelson, Pease and Fred Pearson began measurements in an evacuated pipe about a mile long at the Irvine Ranch near Santa Ana, with the light folded by mirrors into a path of several miles. Michelson died with only part of the programme done. The published result, in 1935, was 299 774 ± 11 km/s, about 18 km/s too low, a reminder that the quoted error bars of that era were often too small.

Chicago and the Nobel Prize, 1890–1931

After two years at Clark University, Michelson accepted in 1892 the post of professor and first head of the department of physics at the new University of Chicago, founded with John D. Rockefeller’s money. The Ryerson Physical Laboratory was dedicated in 1894. He held the post until 1929; in 1925 he was named one of the university’s first Distinguished Service Professors. Robert Millikan, who joined the department in 1896, later wrote the National Academy’s memoir of him and described a man who preferred the laboratory to administration and set the highest standards of precision.

His first marriage ended in divorce in the late 1890s. In 1899 he married Edna Stanton, with whom he had three daughters. He played the violin, tennis and billiards, and he painted landscapes and seascapes in watercolour and oil with real skill. His popular lectures of 1899 became Light Waves and Their Uses (1903). In 1898, with Samuel Stratton, he built a harmonic analyzer that could add or decompose Fourier series of up to 80 terms; its plots of the partial sums of a square wave were part of the background to the discussion of what is now called the Gibbs phenomenon.

In 1907 he received the Nobel Prize in Physics for his optical precision instruments and the spectroscopic and metrological investigations carried out with them, the first Nobel Prize in a science awarded to an American. The same year the Royal Society gave him its Copley Medal. Other honours included the Matteucci Medal, the Elliott Cresson Medal (1912), the Henry Draper Medal (1916), the Gold Medal of the Royal Astronomical Society and the Franklin Medal (both 1923), and the Duddell Medal (1929). He was president of the American Physical Society (1900), of the American Association for the Advancement of Science (1910–11) and of the National Academy of Sciences (1923–27), and a Foreign Member of the Royal Society from 1902.

In the First World War he rejoined the Navy as a reserve officer and designed an optical rangefinder that the Navy adopted. In 1929 he moved to Pasadena to be near Mount Wilson. In 1928 he attended a conference there on the Michelson–Morley experiment, his last major public appearance. He was still at work on the vacuum-pipe measurement when he died in Pasadena on 9 May 1931, aged 78.

Legacy

Michelson’s career shows what precision can do. He rarely proposed new theories; he built instruments that measured an existing quantity a hundred times better, and the improvement repeatedly changed physics.

• Relativity. The null result of 1887 led to the contraction hypothesis and became the most cited experimental support for special relativity. Its modern successors test Lorentz invariance at levels beyond \(10^{-17}\).
• The speed of light. From 1879 to 1926 his values of \(c\) were the most accurate of their time. Since 1983 \(c\) has been fixed at 299 792 458 m/s and the metre is defined from it.
• Metrology. His measurement of the metre in wavelengths of cadmium light led to the krypton definition of 1960. Laser interferometers measuring displacement by counting fringes are now standard in industry.
• Spectroscopy. Visibility curves became Fourier-transform spectroscopy, used from infrared astronomy to chemical analysis.
• Astronomy. The stellar interferometer led to the optical arrays that now image the surfaces of stars, and radio interferometry follows the same principle.
• Gravitational waves. LIGO and Virgo are Michelson interferometers with arms of 4 km and 3 km. In 2015 LIGO detected a change in arm length a small fraction of a proton’s diameter.

A crater on the Moon, the Michelson Laboratory at the Naval Air Weapons Station China Lake, and Michelson Hall at the Naval Academy bear his name.

Sources and further reading

Biography

1. R. A. Millikan, Biographical memoir of Albert Abraham Michelson, National Academy of Sciences, 1938.
2. Albert A. Michelson, biographical, NobelPrize.org.
3. Albert A. Michelson, Wikipedia.
4. D. M. Livingston, The Master of Light: A Biography of Albert A. Michelson, Scribner, 1973 (by his daughter).
5. L. S. Swenson, The Ethereal Aether: A History of the Michelson–Morley–Miller Aether-Drift Experiments, 1880–1930, University of Texas Press, 1972.
6. U.S. Naval Academy, Nimitz Library, Albert Michelson collection.

Physics

1. A. A. Michelson, Experimental Determination of the Velocity of Light (1880), Project Gutenberg.
2. A. A. Michelson, The relative motion of the Earth and the luminiferous ether, American Journal of Science 22 (1881).
3. A. A. Michelson and E. W. Morley, On the relative motion of the Earth and the luminiferous ether, American Journal of Science 34 (1887).
4. A. A. Michelson and E. W. Morley, Influence of motion of the medium on the velocity of light, American Journal of Science 31 (1886).
5. A. A. Michelson and F. G. Pease, Measurement of the diameter of α Orionis with the interferometer, Astrophysical Journal 53 (1921).
6. A. A. Michelson, H. G. Gale and F. Pearson, The effect of the Earth’s rotation on the velocity of light, Astrophysical Journal 61 (1925).
7. A. A. Michelson, Measurement of the velocity of light between Mount Wilson and Mount San Antonio, Astrophysical Journal 65 (1927).
8. A. A. Michelson, Light Waves and Their Uses, University of Chicago Press, 1903; Studies in Optics, 1927.
9. Michelson–Morley experiment and Michelson–Gale–Pearson experiment, Wikipedia.
10. D. Kleppner, Master Michelson’s measurement, Physics Today 60 (2007).
11. M. Born and E. Wolf, Principles of Optics, Cambridge University Press (visibility, coherence, stellar interferometry).