# Chapter 15 Light in Free Space

How do you measure velocity without a clock? As long as you measure something traveling a certain distance, in a certain time, you always need some sort of clock. You can measure the speed of sound simply by watching and listening to the same event a known distance away, just as you can estimate the distance to a thunderstorm by the time difference between lightning and the thunder. But this needs a clock.

On the other hand, you could measure the temperature of the air, and from that estimate the ratio of the pressure to the density of the air. The sound speed is also given by the formula ${{v}_{s}}=\sqrt{{\scriptstyle{}^{P}\!\!\diagup\!\!{}_{{{\rho }_{m}}}\;}}$, where $P$ is the pressure and $\rho_m$ is the density of air. You do not need a clock to measure either the pressure or density of the air. Thus, you can measure the speed of sound either kinematically or by measuring the fluid properties of air.

When Maxwell showed a similar relationship between the permittivity $\varepsilon$ and the permeability $\mu$ and the speed of light:
${{v}_{\text{light}}}={\scriptstyle{}^{1}\!\!\diagup\!\!{}_{\sqrt{\mu \varepsilon }}\;}$,
the situation seemed resolved once and for all. Maxwell’s result seemed to confirm the idea that there must be an all-pervasive medium through which light travels. Thus, many scientists reasoned, the speed of light in free space is really the speed that light travels with respect to the aether. But, what if light does not require a medium to propagate? How then could we either measure the speed of light using a clock, or not using a clock, without running afoul of Galileo’s principle of relativity? On the one hand, if we put our laboratory into motion and should not change, but on the other hand velocities measured with clocks would.

A primary theme of ours, up to this point, has been the impressive success, and ultimate downfall, of the Galilean transformation laws that appear to be a necessary consequence of the principle of relativity. The arbitrary choice of inertial reference frame was employed successfully in both Newtonian mechanics, and electrodynamics, until scientists ran into contradictory results. The spectacular failure of the Galilean transformations forced us to consider an ad hoc modification of the transformation laws between inertial reference frames.

In order to make these connections in the first place, however, the speed of light had to be measured. Light travels so fast that we are tempted to ask, how could anyone without a very accurate modern clock ever measure it? Yet it was first accurately measured in 1676, long before the invention of nanosecond clocks! The ideas, though not the actual measurement, originated with Galileo.

He argued in his 1628 Dialogues that light travels at a finite speed, and was the earliest scientist on record to suggest a measurement for the speed of light.  He proposed that two people with lanterns, standing at a great distance apart, could measure the time required to signal one another by having one person uncover their lantern and the other uncovering their lantern in response.  That way, they would not need a pair of synchronized clocks—one clock would do.

Despite practicing to minimize delay due to reaction time, this method did not show light to take a measurable time to travel.[1] Thus, light must travel faster than the distance of the lanterns divided by human perception time, or faster than about 100 miles per second. Galileo did, however, discover a clock in space—Jupiter’s four large moons. While Galileo named them after his patron, they were renamed Io, Europa, Ganymede, and Callisto, after various love interests of the Greek god Zeus (a.k.a. Jupiter).

Working at the Paris observatory, the Danish astronomer Ole Rømer made detailed measurements of the orbits of the Galilean moons over a number of years, recording the time each moon disappears behind Jupiter, and reemerges on the other side. The time from one disappearance to the next takes Io about 42 hours, and double this for Europa, and double again for Ganymede. Callisto’s period is not an even factor of two greater than Ganymede, but it is not too far off. This led to the first measurement of the speed of light in 1676.

Io, however, does not appear to orbit at exactly the same rate, but rather it appears to orbit faster when the Earth approaches Jupiter, and slower when receding. Rømer knew that the true orbital period of Io should have nothing to do with the relative positions of the Earth and Jupiter, and he therefore concluded that this must be due to the finite speed of light.

Rømer estimated that when the Earth is nearest to Jupiter, eclipses of Io should occur about twenty-two minutes earlier compared to when Earth was farthest away. So, Rømer estimated that light takes about 11 minutes to travel the radius of the earth’s orbit, which was refined, over the years, to the 8.3 minutes per astronomical unit we are familiar with today. This speed was confirmed in 1727, when James Bradley calculated the speed of light using a different astronomical method, that of stellar aberration.[2]

Bradley was not trying to measure stellar aberration, but rather attempting to triangulate the distance from the earth to a star in the constellation Draco by measuring its annual apparent shift in the sky. Bradley did measure such an angular shift, but with a totally different functional dependence than he expected. The shift depended on the perpendicular velocity, rather than the perpendicular position, of the earth compared to the star. Bradley also found that all stars showed the same effect, so the shift must be a property of the light rather than the star.

On a calm rainy day, raindrops fall straight down. However, from the perspective of a moving car, the rain appears to fall at an angle. The faster you travel, the more slanted the rain appears. The angle of the rain only depends on the ratio of your horizontal velocity to the terminal velocity of the raindrops. If you were to look at the car’s speedometer, and measure the angle of the rain from the side window, you could easily calculate the terminal velocity of a raindrop.

The same idea of falling rain applies to stellar aberration. Since Bradley knew the speed of the earth’s orbit around the sun, and the angle of stellar aberration, he could calculate the speed light travels. This, for all practical purposes, established the modern value of 7.2 astronomical units per hour for the speed of light.

There was a catch. The distance from the earth to the sun was still very uncertain. While Bradley could accurately compare measurements of the speed of light, he could not express it in standard units, such as miles per hour, to anywhere near the same accuracy. In fact, Bradley himself had made the best measurement of the astronomical unit, just a few years earlier, with an uncertainty of about 30%. For the next 120 years, each improved measure of the astronomical unit resulted in a corresponding improvement in the value of the speed of light.

Bradley’s discovery of stellar aberration not only corroborated Rømer’s measurement of the speed of light, but also greatly supported Isaac Newton’s particle model, over the wave model of Robert Hooke[3] and Christiaan Huygens.[4] Bradley could easily explain his result using Newton’s light corpuscles, because simple vector addition of the velocities of earth and a light particle gave the correct angles. [5]

On the other hand, the Hooke-Huygens wave theory predicted that the angle would depend on not only the velocities of light and Earth, but also on the velocity of the medium through which light supposedly traveled. Just as our raindrop analogy requires it to be a calm day, the wave model could only explain stellar aberration if—by extraordinary coincidence—the medium were stationary with respect to the sun. Thus, Newton’s corpuscular model dominated until Young’s double slit experiment (p. 544), and Fresnel’s transverse wave theory of light.

Bradley’s observation was the hardest for Fresnel to explain. Perhaps, Fresnel thought, that the aether becomes somewhat dragged along in a transparent medium with refractive index $n$.  If the velocity of the aether then changed by a factor of $\left( 1-\tfrac{1}{{{n}^{2}}} \right),$ Bradley’s observation would become also consistent with a wave model of light.

Fresnel’s hypothesis made a specific prediction: if measurements were made with a water-filled telescope the aberration would be unaffected by the presence of the water. There were no water filled telescopes at the time, but by mid-century the speed of light could finally be measured in the laboratory allowing for further testing of Fresnel’s formula.

The first laboratory measurement of the speed of light was done independently by two French physicists: A. Hippolyte L. Fizeau and Jean Bernard Léon Foucault, whose results Maxwell also cited (p. 607).

The apparatus designed by Fizeau consisted of a light source and a rotating toothed wheel.  The rotating wheel had gaps between the teeth, or gears, through which light could pass. The light was sent out through one place in the teeth, reflected by a mirror placed 8 km away, and if the wheel was rotating fast enough, returned through a different gap.  The speed of light could subsequently be calculated using the distance from mirror to wheel, the speed of wheel rotation, and the spacing between the teeth of the wheel.  The results of Fizeau’s experiment allowed him to calculate a value of 313,300 km/s for the speed of light.[6]

Foucault altered Fizeau’s method by replacing the rotating toothed wheel with a rotating mirror. A light source was then shone onto the rotating mirror where it was reflected onto a distant concave fixed mirror.  The light then reflected from the fixed mirror back to the rotating mirror, and returned to the light source.  If the rotating mirror was spinning at high speeds, the returning light hit it at a slightly different place causing the returning beam to be shifted from its original path.  The speed of light could then be measured by taking into account the speed of mirror rotation, angle of the shift, and the distance between the rotating and fixed mirror.

Foucault used a small steam turbine to spin a mirror at the rate of 800 rotations per second, which he calibrated using a tuning fork and train whistle.[7] A light beam was reflected from it to another mirror 9 meters away. When it returned, 60 nanoseconds later, the mirror had rotated a small amount, causing the return beam to be deflected a little below the source. When the mirror is at any other angle, the light beam is reflected elsewhere in the room and lost. Foucault continually increased the accuracy of this method over the years, and his final measurement in 1862 determined that light travels at 299,796 km/s.[8]

To test the prediction of Newton’s corpuscular theory that light should move faster in water than in air, Foucault introduced a tube of water, 3 meters in length, between the rotating and fixed mirrors. If Newton’s prediction were correct, and the light speed in water is greater than in air, then the return light beam should arrive in less than 60 nanoseconds, and its path would be deflected closer to the source. Instead Foucault found that by introducing the water-filled tube, the light path deflected farther from the source. This showed that light travels more slowly in water than in air, in complete disagreement with the prediction of Newton’s particle model. Foucault’s experiment convinced the majority of scientists that Newton’s theory had to be abandoned.

Their most astonishing experiment, however, was Fizeau’s running water experiment of 1851. Fizeau set up two glass tubes, each about 5 mm in diameter and 1.5 m in length. He then circulated water through them at a speed of about 7 m/s in opposite directions.

Fizeau then built an interferometer by splitting a beam of light, and sending the rays down the two tubes, so one parallel ray traveled upstream and the other downstream.   When the light rays were recombined, Fizeau could detect interference fringe shifts as a function of water velocity, in complete agreement with Fresnel’s aether drag hypothesis. Fizeau concludes his paper thus:

The success of the experiment seems to me to render the adoption of Fresnel’s hypothesis necessary, or at least the law which he found for the expression of the alteration of the velocity of light by the effect of motion of a body; for although that law being found true may be a very strong proof in favour of the hypothesis of which it is only a consequence, perhaps the conception of Fresnel may appear so extraordinary, and in some respects so difficult, to admit, that other proofs and a profound examination on the part of geometricians will still be necessary before adopting it as an expression of the real facts of the case. — Competes Rendus, Sept. 29, 1851.[9]

So, the hypothesis of Fresnel became well-established, but it still seemed too neat a coincidence for Fresnel’s factor, or drag coefficient, to miraculously cancel the effect of motion through the aether. Scientists wondered if they could measure the velocity of the earth relative to the aether. But how?

James Clerk Maxwell turned to astronomy for a method to determine the solar system’s possible motion through the aether by considering observations of the eclipses of Jupiter’s moons, similar to those Rømer used to calculate the speed of light. By comparing the velocity of light when Jupiter is seen from the earth at nearly opposite points of the ecliptic, Maxwell believed that the velocity of the solar system relative to the aether could be found. He wrote that:

The only practicable method of determining directly the relative velocity of the aether with respect to the solar system is to compare the values of the velocity of light deduced from the observation of the eclipses of Jupiter’s satellites when Jupiter is seen from the earth at nearly opposite points of the ecliptic.[10]

Maxwell’s method would measure an effect that is of first order in the velocity of the earth (or solar system) with respect to the aether. That is, the time delay he sought is approximately given by:
$\Delta t=\frac{{{d}_{Earth-Sun}}}{(c-v)}-\frac{{{d}_{Earth-Sun}}}{(c+v)}=\frac{2({{d}_{Earth-Sun}})v}{{{c}^{2}}-{{v}^{2}}}\approx \frac{2({{d}_{Earth-Sun}})v}{{{c}^{2}}}=\frac{2v}{c}{{t}_{0}}$,
where $\left( {{d}_{Earth-Sun}} \right)$ is the diameter of earth’s orbit, $v$ is the speed of the earth through the aether, and $t_0$ is the light-crossing time required for this distance (approximately 16 minutes). So, if $v=30\ \text{km/s}$, for example, one would expect a time delay of about $\Delta t\approx 0.2\ \text{s}$. Unfortunately, it was impossible to detect such a short time delay over the course of half a year.[11]

Sir George Airy, in 1871, attempted to measure the velocity of this aethereal wind, by repeating Bradley’s aberration measurement of the star γ-Draconis, but using a water-filled telescope (shown).[12] So, how would this detect the motion of the Earth?

Since, light moves slower in a medium by a factor of 1/n, where n is the medium’s index of refraction. If the light from a star moves through a telescope filled with water instead of air, it will take n times longer for the light to travel the length of the telescope.

When filled with water, the telescope should, therefore, be tilted more from the vertical to keep the star in view. According to a simple analysis using Snell’s law, the difference between the aberration angles measured under these two different circumstances ought to equal $\left( {{n}^{2}}-1 \right)v/c$, where, again, $v$ is the speed of the earth through the aether.

Airy failed to measure any velocity change, even after observing over the course of two years. The aberration angle was the same whether the telescope was filled with air or water. Instead, Fresnel’s hypothesis of a partial drag of the light by the water itself appeared to explain perfectly Airy’s null result.

Maxwell then made another point about the laboratory experiments that measured the speed of light. In each of these, such as those of Fizeau and Foucault, light beams retrace their paths. So any velocity $v$ the earth may have relative to the aether would affect the time of this round trip by an amount of second order in $v$.

To see this, consider the problem a riverboat traveling upstream to a destination, and downstream on the return trip. The total round trip will still take longer than if on a calm lake, and if L is the one way distance, then the total round trip travel time for the light would be:
$t=\frac{L}{{{v}_{\text{ship}}}+{{v}_{\text{water}}}}+\frac{L}{{{v}_{\text{ship}}}-{{v}_{\text{water}}}}=\frac{2\,L\,{{v}_{\text{ship}}}}{v_{\text{ship}}^{2}-v_{\text{water}}^{2}}\approx \frac{2L}{{{v}_{\text{ship}}}}\left( 1+\frac{v_{\text{water}}^{2}}{v_{\text{ship}}^{2}} \right)$.
In our analogy, the light is the ship and the aether is the river, so:
$\Delta t=t-{{t}_{0}}\approx \frac{2L}{c}\left( 1+\frac{{{v}^{2}}}{{{c}^{2}}} \right)-\frac{2L}{c}\approx \frac{2L}{c}\frac{{{v}^{2}}}{{{c}^{2}}}\approx \frac{{{v}^{2}}}{{{c}^{2}}}{{t}_{0}}$,
where $t_0$ is the time required for light to make the round trip with no wind. If, for example, the speed of the Earth relative to the aether were on the order of its orbital speed about the Sun, then ${{v}^{2}}/{{c}^{2}}\approx {{10}^{-8}}$.   Maxwell noted that the small size of the time delay would be extremely difficult to detect, at least with the technology of his day.

This led Albert A. Michelson to build his perpendicular interferometer, which had produced interference fringes between perpendicular beams of light. As the apparatus rotated, leading him to the too bold conclusion that there was no aether.

Michelson then joined forces with Edward Morley, and in 1887 they again attempted to measure the fringe shift to within one part in 200. Once again, the expected shift was not detected.[13]  Even more than Airy’s observation before it, this null result took nearly every scientist by surprise.

The Irish physicist George Francis FitzGerald, using a recent result of Oliver Heaviside’s, suggested an explanation. Heaviside had shown that the electric field of a charged sphere moving uniformly at speed $v$ is compressed by a factor of $1/\sqrt{1-{{v}^{2}}/{{c}^{2}}}$ in the direction of its motion.[14] Since it seemed likely that electromagnetic forces between molecules hold material bodies together, FitzGerald reasoned, would it be possible that all bodies suffer the same contraction when moving through the aether?[15]

FitzGerald’s hypothesis drew scant attention until Lorentz discovered it independently in 1892.[16] Lorentz then made a straightforward calculation to show that the results of the Michelson-Morley aether-wind experiments would be explained if the length of the interferometer arm were contracted in its direction of motion by this same factor.

Yet, this explanation of Michelson and Morley’s null result appeared to be even more ad hoc than using Fresnel’s theory of partial drag to explain Airy’s stellar aberration experiment. After all, Fizeau’s laboratory measurements had seemingly confirmed Fresnel’s hypothesis long before Airy filled his telescope with water. The nature of molecular forces, on the other hand, was not a settled question at the time. Moreover, since all bodies would be subject to this effect, it was difficult to see how the contraction could be measured. At the close of the nineteenth century, scientists’ understanding of electrodynamics was therefore hampered by confusion regarding the optics of moving bodies.

[1] Foschi, R. and Leone, M, “Galileo, measurement of the velocity of light, and the reaction times”, Perception 38, no. 8 (2009), 1251-9.

[2] James Bradley, “An account of a new discovered motion of the fixed stars”, Phil Trans Roy Soc. 35 (1727), 637-61.

[3] R. Hooke, Micrographia: Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses with Observations and Inquiries Thereupon, (1665), London, Printed by Jo. Martyn, and Ja. Allestry, Printers to the Royal Society, Project Gutenberg E-book number 15491.

[4] C. Christian Huygens, Treatise on Light, translated by S. P. Thomson, (Chicago: University of Chicago Press, 1912).

[5] James Bradley, “An account of a new discovered motion of the fixed stars,” Phil Trans Roy Soc. 35 (1727), 646-9.

[6] Fizeau, “Sur un expérience relative á la vitesse de propagation de la lumière”, Comptes Rendus, (1849), 29 (1849), p. 90. An English translation is available in W.F. Magie, Source Book in Physics, (Cambridge: Harvard University Press, 1935), 341-42.

[7] Foucault’s speed of light experiment in on display at the Musée des arts et métiers in Paris.

[8] Foucault, “Détermination expérimentale de la vitesse e la lumière; parallaxe du Soleil,” Comptes Rendus, 55 (1862), 501-3, and 792-96. English translation is available in W.F. Magie, Source Book in Physics, (Cambridge: Harvard University Press, 1935) 343-44.

[9] Hippolyte Fizeau, “The Hypotheses Relating To The Luminous Æther, And An Experiment Which Appears To Demonstrate That The Motion Of Bodies Alters The Velocity With Which Light Propagates Itself In Their Interior,” Philosophical Magazine, Series 4, vol. 2 (1851), 568-573.

[10] J.C. Maxwell, “Ether,”Encyclopedia Britannica, Ninth Edition 8: 568–572. By carefully reading Maxwell’s quote above, we can derive the expression for by considering the effect of measuring light from Jupiter 6 years apart (the period of Jupiter’s orbit is 12 Earth years) and assuming the velocity to be directed along the line joining Earth and Jupiter at those antipodal points.

[11] J.C. Maxwell, “On a Possible Mode of Detecting a Motion of the Solar System through the Luminiferous Ether,” Nature 21 (1880), 314-315.

[12] G.B. Airy, “On the Supposed Alteration in the Amount of Astronomical Aberration of Light, Produced by the Passage of Light through a Considerable Thickness of Refracting Medium,” Proceedings of the Royal Society of London 20 (1871), 35-39.

[13] A.A. Michelson, “The Relative Motion of the Earth and the Luminiferous Ether,” Am. J. Sci., 122 (1881), 120-129; A.A. Michelson and E.W. Morley, “On the Relative Motion of the Earth and the Luminiferous Ether” Am. J. Sci., 134 (1887), 333-345.

[14] Oliver Heaviside, “On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric,” Philosophical Magazine, Series 5, vol. 27 (167) (1889), 324-339.

[15] G.F. Fitzgerald, “The Ether and Earth’s Atmosphere,” Science, 13 (1889), 390.

[16] H.A. Lorentz, ‘The Relative Motion of the Earth and the Aether,” Amsterdam, Zittingsverlag Akad. v. Wet., 1 (1892), 74.