There are sometimes individuals who seem always to find themselves at the focal points of their times. The physicist George Uhlenbeck was one of these individuals, showing up at all the right times in all the right places at the dawn of modern physics in the 1920’s and 1930’s. He studied under Ehrenfest and Bohr and Born, and he was friends with Fermi and Oppenheimer and Oskar Klein. He taught physics at the universities at Leiden, Michigan, Utrecht, Columbia, MIT and Rockefeller. He was a wide-ranging theoretical physicist who worked on Brownian motion, early string theory, quantum tunneling, and the master equation. Yet he is most famous for the very first thing he did as a graduate student—the discovery of the quantum spin of the electron.
Electron Spin
G. E. Uhlenbeck, and S. Goudsmit, “Spinning electrons and the structure of spectra,” Nature 117, 264-265 (1926).
George Uhlenbeck (1900 – 1988) was born in the Dutch East Indies, the son of a family with a long history in the Dutch military [1]. After the father retired to The Hague, George was expected to follow the family tradition into the military, but he stumbled onto a copy of H. Lorentz’ introductory physics textbook and was hooked. Unfortunately, to attend university in the Netherlands at that time required knowledge of Greek and Latin, which he lacked, so he entered the Institute of Technology in Delft to study chemical engineering. He found the courses dreary.
Fortunately, he was only a few months into his first semester when the language requirement was dropped, and he immediately transferred to the University of Leiden to study physics. He tried to read Boltzmann, but found him opaque, but then read the famous encyclopedia article by the husband and wife team of Paul and Tatiana Ehrenfest on statistical mechanics (see my Physics Today article [2]), which became his lifelong focus.
After graduating, he continued into graduate school, taking classes from Ehrenfest, but lacking funds, he supported himself by teaching classes at a girls high school, until he heard of a job tutoring the son of the Dutch ambassador to Italy. He was off to Rome for three years, where he met Enrico Fermi and took classes from Tullio Bevi-Cevita and Vito Volterra.
However, he nearly lost his way. Surrounded by the rich cultural treasures of Rome, he became deeply interested in art and was seriously considering giving up physics and pursuing a degree in art history. When Ehrenfest got wind of this change in heart, he recalled Uhlenbeck in 1925 to the Netherlands and shrewdly paired him up with another graduate student, Samuel Goudsmit, to work on a new idea proposed by Wolfgang Pauli a few months earlier on the exclusion principle.
Pauli had explained the filling of the energy levels of atoms by introducing a new quantum number that had two values. Once an energy level was filled by two electrons, each carrying one of the two quantum numbers, this energy level “excluded” any further filling by other electrons.
To Uhlenbeck, these two quantum numbers seemed as if they must arise from some internal degree of freedom, and in a flash of insight he imagined that it might be caused if the electron were spinning. Since spin was a form of angular momentum, the spin degree of freedom would combine with orbital angular momentum to produce a composite angular momentum for the quantum levels of atoms.
The idea of electron spin was not immediately embraced by the broader community, and Bohr and Heisenberg and Pauli had their reservations. Fortunately, they all were traveling together to attend the 50th anniversary of Lorentz’ doctoral examination and were met at the train station in Leiden by Ehrenfest and Einstein. As usual, Einstein had grasped the essence of the new physics and explained how the moving electron feels an induced magnetic field which would act on the magnetic moment of the electron to produce spin-orbit coupling. With that, Bohr was convinced.
Uhlenbeck and Goudsmit wrote up their theory in a short article in Nature, followed by a short note by Bohr. A few months later, L. H. Thomas, while visiting Bohr in Copenhagen, explained the factor of two that appears in (what later came to be called) Thomas precession of the electron, cementing the theory of electron spin in the new quantum mechanics.
5-Dimensional Quantum Mechanics
P. Ehrenfest, and G. E. Uhlenbeck, “Graphical illustration of De Broglie’s phase waves in the five-dimensional world of O Klein,” Zeitschrift Fur Physik 39, 495-498 (1926).
Around this time, the Swedish physicist Oskar Klein visited Leiden after returning from three years at the University of Michigan where he had taken advantage of the isolation to develop a quantum theory of 5-dimensional spacetime. This was one of the first steps towards a grand unification of the forces of nature since there was initial hope that gravity and electromagnetism might both be expressed in terms of the five-dimensional space.
An unusual feature of Klein’s 5-dimensional relativity theory was the compactness of the fifth dimension, in which it was “rolled up” into a kind of high-dimensional string with a tiny radius. If the 4-dimensional theory of spacetime was sometimes hard to visualize, here was an even tougher problem.
Uhlenbeck and Ehrenfest met often with Klein during his stay in Leiden, discussing the geometry and consequences of the 5-dimensional theory. Ehrenfest was always trying to get at the essence of physical phenomena in the simplest terms. His famous refrain was “Was ist der Witz?” (What is the point?) [1]. These discussions led to a simple paper in Zeitschrift für Physik published later that year in 1926 by Ehrenfest and Uhlenbeck with the compelling title “Graphical Illustration of De Broglie’s Phase Waves in the Five-Dimensional World of O Klein”. The paper provided the first visualization of the 5-dimensional spacetime with the compact dimension. The string-like character of the spacetime was one of the first forays into modern day “string theory” whose dimensions have now expanded to 11 from 5.
During his visit, Klein also told Uhlenbeck about the relativistic Schrödinger equation that he was working on, which would later become the Klein-Gordon equation. This was a near miss, because what the Klein-Gordon equation was missing was electron spin—which Uhlenbeck himself had introduced into quantum theory—but it would take a few more years before Dirac showed how to incorporate spin into the theory.
Brownian Motion
G. E. Uhlenbeck and L. S. Ornstein, “On the theory of the Brownian motion,” Physical Review 36, 0823-0841 (1930).
After spending time with Bohr in Copenhagen while finishing his PhD, Uhlenbeck visited Max Born at Göttingen where he met J. Robert Oppenheimer who was also visiting Born at that time. When Uhlenbeck traveled to the United States in late summer of 1927 to take a position at the University of Michigan, he was met at the dock in New York by Oppenheimer.
Uhlenbeck was a professor of physics at Michigan for eight years from 1927 to 1935, and he instituted a series of Summer Schools [3] in theoretical physics that attracted international participants and introduced a new generation of American physicists to the rigors of theory that they previously had to go to Europe to find.
In this way, Uhlenbeck was part of a great shift that occurred in the teaching of graduate-level physics of the 1930’s that brought European expertise to the United States. Just a decade earlier, Oppenheimer had to go to Göttingen to find the kind of education that he needed for graduate studies in physics. Oppenheimer brought the new methods back with him to Berkeley, where he established a strong theory department to match the strong experimental activities of E. O. Lawrence. Now, European physicists too were coming to America, an exodus accelerated by the increasing anti-Semitism in Europe under the rise of fascism.
During this time, one of Uhlenbeck’s collaborators was L. S. Ornstein, the director of the Physical Laboratory at the University of Utrecht and a founding member of the Dutch Physical Society. Uhlenbeck and Ornstein were both interested in the physics of Brownian motion, but wished to establish the phenomenon on a more sound physical basis. Einstein’s famous paper of 1905 on Brownian motion had made several Einstein-style simplifications that stripped the complicated theory to its bare essentials, but had lost some of the details in the process, such as the role of inertia at the microscale.
Uhlenbeck and Ornstein published a paper in 1930 that developed the stochastic theory of Brownian motion, including the effects of particle inertia. The stochastic differential equation (SDE) for velocity is
where γ is viscosity, Γ is a fluctuation coefficient, and dw is a “Wiener process”. The Wiener differential dw has unusual properties such that
Uhlenbeck and Ornstein solived this SDE to yield an average velocity
which decays to zero at long times, and a variance
that asymptotes to a finite value at long times. The fluctuation coefficient is thus given by
for a process with characteristic speed v0. An estimate for the fluctuation coefficient can be obtained by considering the force F on an object of size a
For instance, for intracellular transport [4], the fluctuation coefficient has a rough value of Γ = 2 Hz μm2/sec2.
Quantum Tunneling
D. M. Dennison and G. E. Uhlenbeck, “The two-minima problem and the ammonia molecule,” Physical Review 41, 313-321 (1932).
By the early 1930’s, quantum tunnelling of the electron through classically forbidden regions of potential energy was well established, but electrons did not have a monopoly on quantum effects. Entire atoms—electrons plus nucleus—also have quantum wave functions and can experience regions of classically forbidden potential.
Uhlenbeck, with David Dennison, a fellow physicist at Ann Arbor, Michigan, developed the first quantum theory of molecular tunneling for the molecular configuration of ammonia NH3 that can tunnel between the two equivalent configurations. Their use of the WKB approximation in the paper set the standard for subsequent WKB approaches that would play an important role in the calculation of nuclear decay rates.
Master Equation
A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I. The furry model and the fluctuation problem,” Physica 7, 344-360 (1940)
In 1935, Uhlenbeck left Michigan to take up the physics chair recently vacated by Kramers at Utrecht. However, watching the rising Nazism in Europe, he decided to return to the United States, beginning as a visiting professor at Columbia University in New York in 1940. During his visit, he worked with W. E. Lamb and A. Nordsieck on the problem of cosmic ray showers.
Their publication on the topic included a rate equation that is encountered in a wide range of physical phenomena. They called it the “Master Equation” for ease of reference in later parts of the paper, but this phrase stuck, and the “Master Equation” is now a standard tool used by physicists when considering the balances among multiples transitions.
Uhlenbeck never returned to Europe, moving among Michigan, MIT, Princeton and finally settling at Rockefeller University in New York from where he retired in 1971.
Selected Works by George Uhlenbeck:
G. E. Uhlenbeck, and S. Goudsmit, “Spinning electrons and the structure of spectra,” Nature 117, 264-265 (1926).
P. Ehrenfest, and G. E. Uhlenbeck, “On the connection of different methods of solution of the wave equation in multi dimensional spaces,” Proceedings of the Koninklijke Akademie Van Wetenschappen Te Amsterdam 29, 1280-1285 (1926).
P. Ehrenfest, and G. E. Uhlenbeck, “Graphical illustration of De Broglie’s phase waves in the five-dimensional world of O Klein,” Zeitschrift Fur Physik 39, 495-498 (1926).
G. E. Uhlenbeck, and L. S. Ornstein, “On the theory of the Brownian motion,” Physical Review 36, 0823-0841 (1930).
D. M. Dennison, and G. E. Uhlenbeck, “The two-minima problem and the ammonia molecule,” Physical Review 41, 313-321 (1932).
E. Fermi, and G. E. Uhlenbeck, “On the recombination of electrons and positrons,” Physical Review 44, 0510-0511 (1933).
A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I The furry model and the fluctuation problem,” Physica 7, 344-360 (1940).
M. C. Wang, and G. E. Uhlenbeck, “On the Theory of the Brownian Motion-II,” Reviews of Modern Physics 17, 323-342 (1945).
G. E. Uhlenbeck, “50 Years of Spin – Personal Reminiscences,” Physics Today 29, 43-48 (1976).
[3] One of these was the famous 1948 Summer School session where Freeman Dyson met Julian Schwinger after spending days on a cross-country road trip with Richard Feynman. Schwinger and Feynman had developed two different approaches to quantum electrodynamics (QED), which Dyson subsequently reconciled when he took up his position later that year at Princeton’s Institute for Advanced Study, helping to launch the wave of QED that spread out over the theoretical physics community.
Chaos seems to rule our world. Weather events, natural disasters, economic volatility, empire building—all these contribute to the complexities that buffet our lives. It is no wonder that ancient man attributed the chaos to the gods or to the fates, infinitely far from anything we can comprehend as cause and effect. Yet there is a balm to soothe our wounds from the slings of life—Chaos Theory—if not to solve our problems, then at least to understand them.
Chaos Theory is the theory of complex systems governed by multiple factors that produce complicated outputs. The power of the theory is its ability recognize when the complicated outputs are not “random”, no matter how complicated they are, but are in fact determined by the inputs. Furthermore, chaos theory finds structures and patterns within the output—like the fractal structures known as “strange attractors”. These patterns not only are not random, but they tell us about the internal mechanics of the system, and they tell us where to look “on average” for the system behavior.
In other words, chaos theory tames the chaos, and we no longer need to blame gods or the fates.
Henri Poincare (1889)
The first glimpse of the inner workings of chaos was made by accident when Henri Poincaré responded to a mathematics competition held in honor of the King of Sweden. The challenge was to prove whether the solar system was absolutely stable, or whether there was a danger that one day the Earth would be flung from its orbit. Poincaré had already been thinking about the stability of dynamical systems so he wrote up his solution to the challenge and sent it in, believing that he had indeed proven that the solar system was stable.
His entry to the competition was the most convincing, so he was awarded the prize and instructed to submit the manuscript for publication. The paper was already at the printers and coming off the presses when Poincaré was asked by the competition organizer to check one last part of the proof which one of the reviewer’s had questioned relating to homoclinic orbits.
Fig. 1 A homoclinic orbit is an orbit in phase space that intersects itself.
To Poincaré’s horror, as he checked his results against the reviewer’s comments, he found that he had made a fundamental error, and in fact the solar system would never be stable. The problem that he had overlooked had to do with the way that orbits can cross above or below each other on successive passes, leading to a tangle of orbital trajectories that crisscrossed each other in a fine mesh. This is known as the “homoclinic tangle”: it was the first glimpse that deterministic systems could lead to unpredictable results. Most importantly, he had developed the first mathematical tools that would be needed to analyze chaotic systems—such as the Poincaré section—but nearly half a century would pass before these tools would be picked up again.
Poincaré paid out of his own pocket for the first printing to be destroyed and for the corrected version of his manuscript to be printed in its place [1]. No-one but the competition organizers and reviewers ever saw his first version. Yet it was when he was correcting his mistake that he stumbled on chaos for the first time, which is what posterity remembers him for. This little episode in the history of physics went undiscovered for a century before being brought to light by Barrow-Green in her 1997 book Poincaré and the Three Body Problem [2].
Fig. 2 Henri Poincaré’s homoclinic tangle from the Standard Map. (The picture on the right is the Poincaré crater on the moon). For more details, see my blog on Poincaré and his Homoclinic Tangle.
Cartwight and Littlewood (1945)
During World War II, self-oscillations and nonlinear dynamics became strategic topics for the war effort in England. High-power magnetrons were driving long-range radar, keeping Britain alert to Luftwaffe bombing raids, and the tricky dynamics of these oscillators could be represented as a driven van der Pol oscillator. These oscillators had been studied in the 1920’s by the Dutch physicist Balthasar van der Pol (1889–1959) when he was completing his PhD thesis at the University of Utrecht on the topic of radio transmission through ionized gases. van der Pol had built a short-wave triode oscillator to perform experiments on radio diffraction to compare with his theoretical calculations of radio transmission. Van der Pol’s triode oscillator was an engineering feat that produced the shortest wavelengths of the day, making van der Pol intimately familiar with the operation of the oscillator, and he proposed a general form of differential equation for the triode oscillator.
Fig. 3 Driven van der Pol oscillator equation.
Research on the radar magnetron led to theoretical work on driven nonlinear oscillators, including the discovery that a driven van der Pol oscillator could break up into wild and intermittent patterns. This “bad” behavior of the oscillator circuit (bad for radar applications) was the first discovery of chaotic behavior in man-made circuits.
These irregular properties of the driven van der Pol equation were studied by Mary- Lucy Cartwright (1990–1998) (the first woman to be elected a fellow of the Royal Society) and John Littlewood (1885–1977) at Cambridge who showed that the coexistence of two periodic solutions implied that discontinuously recurrent motion—in today’s parlance, chaos— could result, which was clearly undesirable for radar applications. The work of Cartwright and Littlewood [3] later inspired the work by Levinson and Smale as they introduced the field of nonlinear dynamics.
Fig. 4 Mary Cartwright
Andrey Kolmogorov (1954)
The passing of the Russian dictator Joseph Stalin provided a long-needed opening for Soviet scientists to travel again to international conferences where they could meet with their western colleagues to exchange ideas. Four Russian mathematicians were allowed to attend the 1954 International Congress of Mathematics (ICM) held in Amsterdam, the Netherlands. One of those was Andrey Nikolaevich Kolmogorov (1903 – 1987) who was asked to give the closing plenary speech. Despite the isolation of Russia during the Soviet years before World War II and later during the Cold War, Kolmogorov was internationally renowned as one of the greatest mathematicians of his day.
By 1954, Kolmogorov’s interests had spread into topics in topology, turbulence and logic, but no one was prepared for the topic of his plenary lecture at the ICM in Amsterdam. Kolmogorov spoke on the dusty old topic of Hamiltonian mechanics. He even apologized at the start for speaking on such an old topic when everyone had expected him to speak on probability theory. Yet, in the length of only half an hour he laid out a bold and brilliant outline to a proof that the three-body problem had an infinity of stable orbits. Furthermore, these stable orbits provided impenetrable barriers to the diffusion of chaotic motion across the full phase space of the mechanical system. The crucial consequences of this short talk were lost on almost everyone who attended as they walked away after the lecture, but Kolmogorov had discovered a deep lattice structure that constrained the chaotic dynamics of the solar system.
Kolmogorov’s approach used a result from number theory that provides a measure of how close an irrational number is to a rational one. This is an important question for orbital dynamics, because whenever the ratio of two orbital periods is a ratio of integers, especially when the integers are small, then the two bodies will be in a state of resonance, which was the fundamental source of chaos in Poincaré’s stability analysis of the three-body problem. After Komogorov had boldly presented his results at the ICM of 1954 [4], what remained was the necessary mathematical proof of Kolmogorov’s daring conjecture. This would be provided by one of his students, V. I. Arnold, a decade later. But before the mathematicians could settle the issue, an atmospheric scientist, using one of the first electronic computers, rediscovered Poincaré’s tangle, this time in a simplified model of the atmosphere.
Edward Lorenz (1963)
In 1960, with the help of a friend at MIT, the atmospheric scientist Edward Lorenz purchased a Royal McBee LGP-30 tabletop computer to make calculation of a simplified model he had derived for the weather. The McBee used 113 of the latest miniature vacuum tubes and also had 1450 of the new solid-state diodes made of semiconductors rather than tubes, which helped reduce the size further, as well as reducing heat generation. The McBee had a clock rate of 120 kHz and operated on 31-bit numbers with a 15 kB memory. Under full load it used 1500 Watts of power to run. But even with a computer in hand, the atmospheric equations needed to be simplified to make the calculations tractable. Lorenz simplified the number of atmospheric equations down to twelve, and he began programming his Royal McBee.
Progress was good, and by 1961, he had completed a large initial numerical study. One day, as he was testing his results, he decided to save time by starting the computations midway by using mid-point results from a previous run as initial conditions. He typed in the three-digit numbers from a paper printout and went down the hall for a cup of coffee. When he returned, he looked at the printout of the twelve variables and was disappointed to find that they were not related to the previous full-time run. He immediately suspected a faulty vacuum tube, as often happened. But as he looked closer at the numbers, he realized that, at first, they tracked very well with the original run, but then began to diverge more and more rapidly until they lost all connection with the first-run numbers. The internal numbers of the McBee had a precision of 6 decimal points, but the printer only printed three to save time and paper. His initial conditions were correct to a part in a thousand, but this small error was magnified exponentially as the solution progressed. When he printed out the full six digits (the resolution limit for the machine), and used these as initial conditions, the original trajectory returned. There was no mistake. The McBee was working perfectly.
At this point, Lorenz recalled that he “became rather excited”. He was looking at a complete breakdown of predictability in atmospheric science. If radically different behavior arose from the smallest errors, then no measurements would ever be accurate enough to be useful for long-range forecasting. At a more fundamental level, this was a break with a long-standing tradition in science and engineering that clung to the belief that small differences produced small effects. What Lorenz had discovered, instead, was that the deterministic solution to his 12 equations was exponentially sensitive to initial conditions (known today as SIC).
The more Lorenz became familiar with the behavior of his equations, the more he felt that the 12-dimensional trajectories had a repeatable shape. He tried to visualize this shape, to get a sense of its character, but it is difficult to visualize things in twelve dimensions, and progress was slow, so he simplified his equations even further to three variables that could be represented in a three-dimensional graph [5].
Fig. 5 Two-dimensional projection of the three-dimensional Lorenz Butterfly.
V. I. Arnold (1964)
Meanwhile, back in Moscow, an energetic and creative young mathematics student knocked on Kolmogorov’s door looking for an advisor for his undergraduate thesis. The youth was Vladimir Igorevich Arnold (1937 – 2010), who showed promise, so Kolmogorov took him on as his advisee. They worked on the surprisingly complex properties of the mapping of a circle onto itself, which Arnold filed as his dissertation in 1959. The circle map holds close similarities with the periodic orbits of the planets, and this problem led Arnold down a path that drew tantalizingly close to Kolmogorov’s conjecture on Hamiltonian stability. Arnold continued in his PhD with Kolmogorov, solving Hilbert’s 13th problem by showing that every function of n variables can be represented by continuous functions of a single variable. Arnold was appointed as an assistant in the Faculty of Mechanics and Mathematics at Moscow State University.
Arnold’s habilitation topic was Kolmogorov’s conjecture, and his approach used the same circle map that had played an important role in solving Hilbert’s 13th problem. Kolmogorov neither encouraged nor discouraged Arnold to tackle his conjecture. Arnold was led to it independently by the similarity of the stability problem with the problem of continuous functions. In reference to his shift to this new topic for his habilitation, Arnold stated “The mysterious interrelations between different branches of mathematics with seemingly no connections are still an enigma for me.” [6]
Arnold began with the problem of attracting and repelling fixed points in the circle map and made a fundamental connection to the theory of invariant properties of action-angle variables . These provided a key element in the proof of Kolmogorov’s conjecture. In late 1961, Arnold submitted his results to the leading Soviet physics journal—which promptly rejected it because he used forbidden terms for the journal, such as “theorem” and “proof”, and he had used obscure terminology that would confuse their usual physicist readership, terminology such as “Lesbesgue measure”, “invariant tori” and “Diophantine conditions”. Arnold withdrew the paper.
Arnold later incorporated an approach pioneered by Jurgen Moser [7] and published a definitive article on the problem of small divisors in 1963 [8]. The combined work of Kolmogorov, Arnold and Moser had finally established the stability of irrational orbits in the three-body problem, the most irrational and hence most stable orbit having the frequency of the golden mean. The term “KAM theory”, using the first initials of the three theorists, was coined in 1968 by B. V. Chirikov, who also introduced in 1969 what has become known as the Chirikov map (also known as the Standard map ) that reduced the abstract circle maps of Arnold and Moser to simple iterated functions that any student can program easily on a computer to explore KAM invariant tori and the onset of Hamiltonian chaos, as in Fig. 1 [9].
Fig. 6 The Chirikov Standard Map when the last stable orbits are about to dissolve for ε = 0.97.
Sephen Smale (1967)
Stephen Smale was at the end of a post-graduate fellowship from the National Science Foundation when he went to Rio to work with Mauricio Peixoto. Smale and Peixoto had met in Princeton in 1960 where Peixoto was working with Solomon Lefschetz (1884 – 1972) who had an interest in oscillators that sustained their oscillations in the absence of a periodic force. For instance, a pendulum clock driven by the steady force of a hanging weight is a self-sustained oscillator. Lefschetz was building on work by the Russian Aleksandr A. Andronov (1901 – 1952) who worked in the secret science city of Gorky in the 1930’s on nonlinear self-oscillations using Poincaré’s first return map. The map converted the continuous trajectories of dynamical systems into discrete numbers, simplifying problems of feedback and control.
The central question of mechanical control systems, even self-oscillating systems, was how to attain stability. By combining approaches of Poincaré and Lyapunov, as well as developing their own techniques, the Gorky school became world leaders in the theory and applications of nonlinear oscillations. Andronov published a seminal textbook in 1937 The Theory of Oscillations with his colleagues Vitt and Khaykin, and Lefschetz had obtained and translated the book into English in 1947, introducing it to the West. When Peixoto returned to Rio, his interest in nonlinear oscillations captured the imagination of Smale even though his main mathematical focus was on problems of topology. On the beach in Rio, Smale had an idea that topology could help prove whether systems had a finite number of periodic points. Peixoto had already proven this for two dimensions, but Smale wanted to find a more general proof for any number of dimensions.
Norman Levinson (1912 – 1975) at MIT became aware of Smale’s interests and sent off a letter to Rio in which he suggested that Smale should look at Levinson’s work on the triode self-oscillator (a van der Pol oscillator), as well as the work of Cartwright and Littlewood who had discovered quasi-periodic behavior hidden within the equations. Smale was puzzled but intrigued by Levinson’s paper that had no drawings or visualization aids, so he started scribbling curves on paper that bent back upon themselves in ways suggested by the van der Pol dynamics. During a visit to Berkeley later that year, he presented his preliminary work, and a colleague suggested that the curves looked like strips that were being stretched and bent into a horseshoe.
Smale latched onto this idea, realizing that the strips were being successively stretched and folded under the repeated transformation of the dynamical equations. Furthermore, because dynamics can move forward in time as well as backwards, there was a sister set of horseshoes that were crossing the original set at right angles. As the dynamics proceeded, these two sets of horseshoes were repeatedly stretched and folded across each other, creating an infinite latticework of intersections that had the properties of the Cantor set. Here was solid proof that Smale’s original conjecture was wrong—the dynamics had an infinite number of periodicities, and they were nested in self-similar patterns in a latticework of points that map out a Cantor-like set of points. In the two-dimensional case, shown in the figure, the fractal dimension of this lattice is D = ln4/ln3 = 1.26, somewhere in dimensionality between a line and a plane. Smale’s infinitely nested set of periodic points was the same tangle of points that Poincaré had noticed while he was correcting his King Otto Prize manuscript. Smale, using modern principles of topology, was finally able to put rigorous mathematical structure to Poincaré’s homoclinic tangle. Coincidentally, Poincaré had launched the modern field of topology, so in a sense he sowed the seeds to the solution to his own problem.
Fig. 7 The horseshoe takes regions of phase space and stretches and folds them over and over to create a lattice of overlapping trajectories.
Ruelle and Takens (1971)
The onset of turbulence was an iconic problem in nonlinear physics with a long history and a long list of famous researchers studying it. As far back as the Renaissance, Leonardo da Vinci had made detailed studies of water cascades, sketching whorls upon whorls in charcoal in his famous notebooks. Heisenberg, oddly, wrote his PhD dissertation on the topic of turbulence even while he was inventing quantum mechanics on the side. Kolmogorov in the 1940’s applied his probabilistic theories to turbulence, and this statistical approach dominated most studies up to the time when David Ruelle and Floris Takens published a paper in 1971 that took a nonlinear dynamics approach to the problem rather than statistical, identifying strange attractors in the nonlinear dynamical Navier-Stokes equations [10]. This paper coined the phrase “strange attractor”. One of the distinct characteristics of their approach was the identification of a bifurcation cascade. A single bifurcation means a sudden splitting of an orbit when a parameter is changed slightly. In contrast, a bifurcation cascade was not just a single Hopf bifurcation, as seen in earlier nonlinear models, but was a succession of Hopf bifurcations that doubled the period each time, so that period-two attractors became period-four attractors, then period-eight and so on, coming fast and faster, until full chaos emerged. A few years later Gollub and Swinney experimentally verified the cascade route to turbulence , publishing their results in 1975 [11].
Fig. 8 Bifurcation cascade of the logistic map.
Feigenbaum (1978)
In 1976, computers were not common research tools, although hand-held calculators now were. One of the most famous of this era was the Hewlett-Packard HP-65, and Feigenbaum pushed it to its limits. He was particularly interested in the bifurcation cascade of the logistic map [12]—the way that bifurcations piled on top of bifurcations in a forking structure that showed increasing detail at increasingly fine scales. Feigenbaum was, after all, a high-energy theorist and had overlapped at Cornell with Kenneth Wilson when he was completing his seminal work on the renormalization group approach to scaling phenomena. Feigenbaum recognized a strong similarity between the bifurcation cascade and the ideas of real-space renormalization where smaller and smaller boxes were used to divide up space.
One of the key steps in the renormalization procedure was the need to identify a ratio of the sizes of smaller structures to larger structures. Feigenbaum began by studying how the bifurcations depended on the increasing growth rate. He calculated the threshold values rm for each of the bifurcations, and then took the ratios of the intervals, comparing the previous interval (rm-1 – rm-2) to the next interval (rm – rm-1). This procedure is like the well-known method to calculate the golden ratio = 1.61803 from the Fibonacci series, and Feigenbaum might have expected the golden ratio to emerge from his analysis of the logistic map. After all, the golden ratio has a scary habit of showing up in physics, just like in the KAM theory. However, as the bifurcation index m increased in Feigenbaum’s study, this ratio settled down to a limiting value of 4.66920. Then he did what anyone would do with an unfamiliar number that emerges from a physical calculation—he tried to see if it was a combination of other fundamental numbers, like pi and Euler’s constant e, and even the golden ratio. But none of these worked. He had found a new number that had universal application to chaos theory [13].
Fig. 9 The ratio of the limits of successive cascades leads to a new universal number (the Feigenbaum number).
Gleick (1987)
By the mid-1980’s, chaos theory was seeping in to a broadening range of research topics that seemed to span the full breadth of science, from biology to astrophysics, from mechanics to chemistry. A particularly active group of chaos practitioners were J. Doyn Farmer, James Crutchfield, Norman Packard and Robert Shaw who founded the Dynamical Systems Collective at the University of California, Santa Cruz. One of the important outcomes of their work was a method to reconstruct the state space of a complex system using only its representative time series [14]. Their work helped proliferate the techniques of chaos theory into the mainstream. Many who started using these techniques were only vaguely aware of its long history until the science writer James Gleick wrote a best-selling history of the subject that brought chaos theory to the forefront of popular science [15]. And the rest, as they say, is history.
References
[1] Poincaré, H. and D. L. Goroff (1993). New methods of celestial mechanics. Edited and introduced by Daniel L. Goroff. New York, American Institute of Physics.
[2] J. Barrow-Green, Poincaré and the three body problem (London Mathematical Society, 1997).
[3] Cartwright,M.L.andJ.E.Littlewood(1945).“Onthenon-lineardifferential equation of the second order. I. The equation y′′ − k(1 – yˆ2)y′ + y = bλk cos(λt + a), k large.” Journal of the London Mathematical Society 20: 180–9. Discussed in Aubin, D. and A. D. Dalmedico (2002). “Writing the History of Dynamical Systems and Chaos: Longue DurÈe and Revolution, Disciplines and Cultures.” Historia Mathematica, 29: 273.
[4] Kolmogorov, A. N., (1954). “On conservation of conditionally periodic motions for a small change in Hamilton’s function.,” Dokl. Akad. Nauk SSSR (N.S.), 98: 527–30.
[5] Lorenz, E. N. (1963). “Deterministic Nonperiodic Flow.” Journal of the Atmo- spheric Sciences 20(2): 130–41.
[6] Arnold,V.I.(1997).“From superpositions to KAM theory,”VladimirIgorevich Arnold. Selected, 60: 727–40.
[7] Moser, J. (1962). “On Invariant Curves of Area-Preserving Mappings of an Annulus.,” Nachr. Akad. Wiss. Göttingen Math.-Phys, Kl. II, 1–20.
[8] Arnold, V. I. (1963). “Small denominators and problems of the stability of motion in classical and celestial mechanics (in Russian),” Usp. Mat. Nauk., 18: 91–192,; Arnold, V. I. (1964). “Instability of Dynamical Systems with Many Degrees of Freedom.” Doklady Akademii Nauk Sssr 156(1): 9.
[9] Chirikov, B. V. (1969). Research concerning the theory of nonlinear resonance andstochasticity. Institute of Nuclear Physics, Novosibirsk. 4. Note: The Standard Map Jn+1 =Jn +εsinθn θn+1 =θn +Jn+1 is plotted in Fig. 3.31 in Nolte, Introduction to Modern Dynamics (2015) on p. 139. For small perturbation ε, two fixed points appear along the line J = 0 corresponding to p/q = 1: one is an elliptical point (with surrounding small orbits) and the other is a hyperbolic point where chaotic behavior is first observed. With increasing perturbation, q elliptical points and q hyperbolic points emerge for orbits with winding numbers p/q with small denominators (1/2, 1/3, 2/3 etc.). Other orbits with larger q are warped by the increasing perturbation but are not chaotic. These orbits reside on invariant tori, known as the KAM tori, that do not disintegrate into chaos at small perturbation. The set of KAM tori is a Cantor-like set with non- zero measure, ensuring that stable behavior can survive in the presence of perturbations, such as perturbation of the Earth’s orbit around the Sun by Jupiter. However, with increasing perturbation, orbits with successively larger values of q disintegrate into chaos. The last orbits to survive in the Standard Map are the golden mean orbits with p/q = φ–1 and p/q = 2–φ. The critical value of the perturbation required for the golden mean orbits to disintegrate into chaos is surprisingly large at εc = 0.97.
[10] Ruelle,D. and F.Takens (1971).“OntheNatureofTurbulence.”Communications in Mathematical Physics 20(3): 167–92.
[11] Gollub, J. P. and H. L. Swinney (1975). “Onset of Turbulence in a Rotating Fluid.” Physical Review Letters, 35(14): 927–30.
[12] May, R. M. (1976). “Simple Mathematical-Models with very complicated Dynamics.” Nature, 261(5560): 459–67.
[13] M. J. Feigenbaum, “Quantitative Universality for a Class of Nnon-linear Transformations,” Journal of Statistical Physics 19, 25-52 (1978).
[14] Packard, N.; Crutchfield, J. P.; Farmer, J. Doyne; Shaw, R. S. (1980). “Geometry from a Time Series”. Physical Review Letters. 45 (9): 712–716.
Albert Michelson was the first American to win a Nobel Prize in science. He was awarded the Nobel Prize in physics in 1907 for the invention of his eponymous interferometer and for its development as a precision tool for metrology. On board ship traveling to Sweden from London to receive his medal, he was insulted by the British author Rudyard Kipling (that year’s Nobel Laureate in literature) who quipped that America was filled with ignorant masses who wouldn’t amount to anything.
Notwithstanding Kipling’s prediction, across the following century, Americans were awarded 96 Nobel prizes in physics. The next closest nationalities were Germany with 28, the United Kingdom with 25 and France with 18. These are ratios of 3:1, 4:1 and 5:1. Why was the United States so dominant, and why was Rudyard Kipling so wrong?
At the same time that American scientists were garnering the lion’s share of Nobel prizes in physics in the 20th century, the American real (inflation-adjusted) gross-domestic-product (GDP) grew from 60 billion dollars to 20 trillion dollars, making up about a third of the world-wide GDP, even though it has only about 5% of the world population. So once again, why was the United States so dominant across the last century? What factors contributed to this success?
The answers are complicated, with many contributing factors and lots of shades of gray. But two factors stand out that grew hand-in-hand over the century; these are:
1) The striking rise of American elite universities, and
2) The significant gain in the US brain trust through immigration
Albert Michelson is a case in point.
The Firestorms of Albert Michelson
Albert Abraham Michelson was, to some, an undesirable immigrant, born poor in Poland to a Jewish family who made the arduous journey through the Panama Canal in the second wave of 49ers swarming over the California gold country. Michelson grew up in the Wild West, first in the rough town of Murphy’s Camp in California, in foothills of the Sierras. After his father’s supply store went up in flames, they moved to Virginia City, Nevada. His younger brother Charlie lived by the gun (after Michelson had left home), providing meat and protection for supply trains during the Apache wars in the Southwest. This was America in the raw.
Yet Michelson was a prodigy. He outgrew the meager educational possibilities in the mining towns, so his family scraped together enough money to send him to a school in San Francisco, where he excelled. Later, in Virginia City, an academic competition was held for a special appointment to the Naval Academy in Annapolis, and Michelson tied for first place, but the appointment went to the other student who was the son of a Civil War Vet.
With the support of the local Jewish community, Michelson took a train to Washington DC (traveling on the newly-completed Transcontinental Railway, passing over the spot where a golden spike had been driven one month prior into a railroad tie made of Californian laurel) to make his case directly. He met with President Grant at the White House, but all the slots at Annapolis had been filled. Undaunted, Michelson camped out for three days in the waiting room of the office of an Annapolis Admiral, who finally relented and allowed Michelson to take the entrance exam. Still, there was no place for him at the Academy.
Discouraged, Michelson bought a ticket and boarded the train for home. One can only imagine his shock when he heard his name called out by a someone walking down the car aisle. It was a courier from the White House. Michelson met again with Grant, who made an extraordinary extra appointment for Michelson at Annapolis; the Admiral had made his case for him. With no time to return home, he was on board ship for his first training cruise within a week, returning a month later to start classes.
Fig. 1 Albert Abraham Michelson
Years later, as Michelson prepared, with Edmund Morley, to perform the most sensitive test ever made of the motion of the Earth, using his recently-invented “Michelson Interferometer”, the building with his lab went up in flames, just like his father’s goods store had done years before. This was a trying time for Michelson. His first marriage was on the rocks, and he had just recovered from having a nervous breakdown (his wife at one point tried to have him committed to an insane asylum from where patients rarely ever returned). Yet with Morley’s help, they completed the measurement.
To Michelson’s dismay, the exquisite experiment with the finest sensitivity—that should have detected a large deviation of the fringes depending on the orientation of the interferometer relative to the motion of the Earth through space—gave a null result. They published their findings, anyway, as one more puzzle in the question of the speed of light, little knowing how profound this “Michelson-Morley” experiment would be in the history of modern physics and the subsequent development of the relativity theory of Albert Einstein (another immigrant).
Putting the disappointing null result behind him, Michelson next turned his ultra-sensitive interferometer to the problem of replacing the platinum meter-bar standard in Paris with a new standard that was much more fundamental—wavelengths of light. This work, unlike his null result, led to practical success for which he was awarded the Nobel Prize in 1907 (not for his null result with Morley).
Michelson’s Nobel Prize in physics in 1907 did not immediately open the floodgates. Sixteen years passed before the next Nobel in physics went to an American (Robert Millikan). But after 1936 (as many exiles from fascism in Europe immigrated to the US) Americans were regularly among the prize winners.
List of American Nobel Prizes in Physics
* (I) designates an immigrant.
1907 Albert Michelson (I) Optical precision instruments and metrology
1923 Robert Millikan Elementary charge and photoelectric effect
1927 Arthur Compton The Compton effect
1936 Carl David Anderson Discovery of the positron
1937 Clinton Davisson Diffraction of electrons by crystals
1939 Ernest Lawrence Invention of the cyclotron
1943 Otto Stern (I) Magnetic moment of the proton
1944 Isidor Isaac Rabi (I) Magnetic properties of atomic nuclei
1946 Percy Bridgman High pressure physics
1952 E. M. Purcell Nuclear magnetic precision measurements
1952 Felix Bloch (I) Nuclear magnetic precision measurements
1955 Willis Lamb Fine structure of the hydrogen spectrum
1955 Polykarp Kusch (I) Magnetic moment of the electron
1956 William Shockley (I) Discovery of the transistor effect
1956 John Bardeen Discovery of the transistor effect
1956 Walter H. Brattain (I) Discovery of the transistor effect
1957 Chen Ning Yang (I) Parity laws of elementary particles
1957 Tsung-Dao Lee (I) Parity laws of elementary particles
1959 Owen Chamberlain Discovery of the antiproton
1959 Emilio Segrè (I) Discovery of the antiproton
1960 Donald Glaser Invention of the bubble chamber
1961 Robert Hofstadter The structure of nucleons
1963 Maria Goeppert-Mayer (I) Nuclear shell structure
1963 Eugene Wigner (I) Fundamental symmetry principles
1964 Charles Townes Quantum electronics
1965 Richard Feynman Quantum electrodynamics
1965 Julian Schwinger Quantum electrodynamics
1967 Hans Bethe (I) Theory of nuclear reactions
1968 Luis Alvarez Hydrogen bubble chamber
1969 Murray Gell-Mann Classification of elementary particles and interactions
1972 John Bardeen Theory of superconductivity
1972 Leon N. Cooper Theory of superconductivity
1972 Robert Schrieffer Theory of superconductivity
1973 Ivar Giaever (I) Tunneling phenomena
1975 Ben Roy Mottelson The structure of the atomic nucleus
1975 James Rainwater The structure of the atomic nucleus
1976 Burton Richter Discovery of a heavy elementary particle
1976 Samuel C. C. Ting Discovery of a heavy elementary particle
1977 Philip Anderson Magnetic and disordered systems
1977 John van Vleck Magnetic and disordered systems
1978 Robert Wilson Discovery of cosmic microwave background radiation
1978 Arno Penzias (I) Discovery of cosmic microwave background radiation
1979 Steven Weinberg Unified weak and electromagnetic interaction
1979 Sheldon Glashow Unified weak and electromagnetic interaction
1980 James Cronin Symmetry principles in the decay of neutral K-mesons
1980 Val Fitch Symmetry principles in the decay of neutral K-mesons
1981 Nicolaas Bloembergen (I) Nonlinear Optics
1981 Arthur Schawlow Development of laser spectroscopy
1982 Kenneth Wilson Theory for critical phenomena and phase transitions
1983 William Fowler Formation of the chemical elements in the universe
1983 Subrahmanyan Chandrasekhar (I) The evolution of the stars
1988 Leon Lederman Discovery of the muon neutrino
1988 Melvin Schwartz Discovery of the muon neutrino
1988 Jack Steinberger (I) Discovery of the muon neutrino
1989 Hans Dehmelt (I) Ion trap
1989 Norman Ramsey Atomic clocks
1990 Jerome Friedman Deep inelastic scattering of electrons on nucleons
1990 Henry Kendall Deep inelastic scattering of electrons on nucleons
1993 Russell Hulse Discovery of a new type of pulsar
1993 Joseph Taylor Jr. Discovery of a new type of pulsar
1994 Clifford Shull Neutron diffraction
1995 Martin Perl Discovery of the tau lepton
1995 Frederick Reines Detection of the neutrino
1996 David Lee Discovery of superfluidity in helium-3
1996 Douglas Osheroff Discovery of superfluidity in helium-3
1996 Robert Richardson Discovery of superfluidity in helium-3
1997 Steven Chu Laser atom traps
1997 William Phillips Laser atom traps
1998 Horst Störmer (I) Fractionally charged quantum Hall effect
1998 Robert Laughlin Fractionally charged quantum Hall effect
1998 Daniel Tsui (I) Fractionally charged quantum Hall effect
2000 Jack Kilby Integrated circuit
2001 Eric Cornell Bose-Einstein condensation
2001 Carl Wieman Bose-Einstein condensation
2002 Raymond Davis Jr. Cosmic neutrinos
2002 Riccardo Giacconi (I) Cosmic X-ray sources
2003 Anthony Leggett (I) The theory of superconductors and superfluids
2003 Alexei Abrikosov (I) The theory of superconductors and superfluids
2004 David Gross Asymptotic freedom in the strong interaction
2004 H. David Politzer Asymptotic freedom in the strong interaction
2004 Frank Wilczek Asymptotic freedom in the strong interaction
2005 John Hall Quantum theory of optical coherence
2005 Roy Glauber Quantum theory of optical coherence
2006 John Mather Anisotropy of the cosmic background radiation
2006 George Smoot Anisotropy of the cosmic background radiation
2008 Yoichiro Nambu (I) Spontaneous broken symmetry in subatomic physics
2009 Willard Boyle (I) CCD sensor
2009 George Smith CCD sensor
2009 Charles Kao (I) Fiber optics
2011 Saul Perlmutter Accelerating expansion of the Universe
2011 Brian Schmidt Accelerating expansion of the Universe
2011 Adam Riess Accelerating expansion of the Universe
2012 David Wineland Atom Optics
2014 Shuji Nakamura (I) Blue light-emitting diodes
2016 F. Duncan Haldane (I) Topological phase transitions
2016 John Kosterlitz (I) Topological phase transitions
2017 Rainer Weiss (I) LIGO detector and gravitational waves
2017 Kip Thorne LIGO detector and gravitational waves
2017 Barry Barish LIGO detector and gravitational waves
(Note: This list does not include Enrico Fermi, who was awarded the Nobel Prize while in Italy. After traveling to Stockholm to receive the award, he did not return to Italy, but went to the US to protect his Jewish wife from the new race laws enacted by the nationalist government of Italy. There are many additional Nobel prize winners not on this list (like Albert Einstein) who received the Nobel Prize while in their own country but who then came to the US to teach at US institutions.)
Immigration and Elite Universities
A look at the data behind the previous list tells a striking story: 1) Nearly all of the American Nobel Prizes in physics were awarded for work performed at elite American universities; 2) Roughly a third of the prizes went to immigrants. And for those prize winners who were not immigrants themselves, many were taught by, or studied under, immigrant professors at those elite universities.
Elite universities are not just the source of Nobel Prizes, but are engines of the economy. The Tech Sector may contribute only 10% of the US GDP, but 85% of our GDP is attributed to “innovation”, much of coming out of our universities. Our “inventive” economy is driving the American standard of living and keeps us competitive in the worldwide market.
Today, elite universities, as well as immigration, are under attack by forces who want to make America great again. Legislatures in some states have passed laws restricting how those universities hire and teach, and more states are following suite. Some new state laws restrict where Chinese-born professors, who are teaching and conducting research at American universities, can or cannot buy houses. And some members of Congress recently ambushed the leaders of a few of our most elite universities (who failed spectacularly to use common sense), using the excuse of a non-academic issue to turn universities into a metaphor for the supposed evils of elitism.
But the forces seeking to make America great again may be undermining the very thing that made America great in the first place.
They want to cook the goose, but they are overlooking the golden eggs.
One hundred years ago this month, in Feb. 1924, a hereditary member of the French nobility, Louis Victor Pierre Raymond, the 7th Duc de Broglie, published a landmark paper in the Philosophical Magazine of London [1] that revolutionized the nascent quantum theory of the day.
Prior to de Broglie’s theory of quantum matter waves, quantum physics had been mired in ad hoc phenomenological prescriptions like Bohr’s theory of the hydrogen atom and Sommerfeld’s theory of adiabatic invariants. After de Broglie, Erwin Schrödinger would turn the concept of matter waves into the theory of wave mechanics that we still practice today.
Fig. 1 The 1924 paper by de Broglie in the Philosophical Magazine.
The story of how de Broglie came to his seminal idea had an odd twist, based on an initial misconception that helped him get the right answer ahead of everyone else, for which he was rewarded with the Nobel Prize in Physics.
de Broglie’s Early Days
When Louis de Broglie was a student, his older brother Maurice (the 6th Duc de Broglie) was already a practicing physicist making important discoveries in x-ray physics. Although Louis initially studied history in preparation for a career in law, and he graduated from the Sorbonne with a degree in history, his brother’s profession drew him like a magnet. He also read Poincaré at this critical juncture in his career, and he was hooked. He enrolled in the Faculty of Sciences for his advanced degree, but World War I side-tracked him into the signal corps, where he was assigned to the wireless station on top of the Eiffel Tower. He may have participated in the famous interception of a coded German transmission in 1918 that helped turn the tide of the war.
Beginning in 1919, Louis began assisting his brother in the well-equiped private laboratory that Maurice had outfitted in the de Broglie ancestral home. At that time Maurice was performing x-ray spectroscopy of the inner quantum states of atoms, and he was struck by the duality of x-ray properties that made them behave like particles under some conditions and like waves in others.
Fig. 2 Maurice de Broglie in his private laboratory (Figure credit).
Through his close work with his brother, Louis also came to subscribe to the wave-particle duality of x-rays and chose the topic for his PhD thesis—and hence the twist that launched de Broglie backwards towards his epic theory.
de Broglie’s Massive Photons
Today, we say that photons have energy and momentum although they are massless. The momentum is a simple consequence of Einstein’s special relativity
And if m = 0, then
and momentum requires energy but not necessarily mass.
But de Broglie started out backwards. He was so convinced of the particle-like nature of the x-ray photons, that he first considered what would happen if the photons actually did have mass. He constructed a massive photon and compared its proper frequency with a Lorentz-boosted frequency observed in a laboratory. The frequency he set for the photon was like an internal clock, set by its rest-mass energy and by Bohr’s quantization condition
He then boosted it into the lab frame by time dilation
But the energy would be transformed according to
with a corresponding frequency
which is in direct contradiction with Bohr’s quantization condition. What is the resolution of this seeming paradox?
de Broglie’s Matter Wave
de Broglie realized that his “massive photon” must satisfy a condition relating the observed lab frequency to the transformed frequency, such that
This only made sense if his “massive photon” could be represented as a wave with a frequency
that propagated with a phase velocity given by c/β. (Note that β < 1 so that the phase velocity is greater than the speed of light, which is allowed as long as it does not transmit any energy.)
To a modern reader, this all sounds alien, but only because this work in early 1924 represented his first pass at his theory. As he worked on this thesis through 1924, finally defending it in November of that year, he refined his arguments, recognizing that when he combined his frequency with his phase velocity,
it yielded the wavelength for a matter wave to be
where p was the relativistic mechanical momentum of a massive particle.
Using this wavelength, he explained Bohr’s quantization condition as a simple standing wave of the matter wave. In the light of this derivation, de Broglie wrote
We are then inclined to admit that any moving body may be accompanied by a wave and that it is impossible to disjoin motion of body and propagation of wave.
pg. 450, Philosophical Magazine of London (1924)
Here was the strongest statement yet of the wave-particle duality of quantum particles. de Broglie went even further and connected the ideas of waves and rays through the Hamilton-Jacobi formalism, an approach that Dirac would extend several years later, establishing the formal connection between Hamiltonian physics and wave mechanics. Furthermore, de Broglie conceived of a “pilot wave” interpretation that removed some of Einstein’s discomfort with the random character of quantum measurement that ultimately led Einstein to battle Bohr in their famous debates, culminating in the iconic EPR paper that has become a cornerstone for modern quantum information science. After the wave-like nature of particles was confirmed in the Davisson-Germer experiments, de Broglie received the Nobel Prize in Physics in 1929.
Fig. 4 A standing matter wave is a stationary state of constructive interference. This wavefunction is in the L = 5 quantum manifold of the hydrogen atom.
Louis de Broglie was clearly ahead of his times. His success was partly due to his isolation from the dogma of the day. He was able to think without the constraints of preconceived ideas. But as soon as he became a regular participant in the theoretical discussions of his day, and bowed under the pressure from Copenhagen, his creativity essentially ceased. The subsequent development of quantum mechanics would be dominated by Heisenberg, Born, Pauli, Bohr and Schrödinger, beginning at the 1927 Solvay Congress held in Brussels.
These days, the physics breakthroughs in the news that really catch the eye tend to be Astro-centric. Partly, this is due to the new data coming from the James Webb Space Telescope, which is the flashiest and newest toy of the year in physics. But also, this is part of a broader trend in physics that we see in the interest statements of physics students applying to graduate school. With the Higgs business winding down for high energy physics, and solid state physics becoming more engineering, the frontiers of physics have pushed to the skies, where there seem to be endless surprises.
To be sure, quantum information physics (a hot topic) and AMO (atomic and molecular optics) are performing herculean feats in the laboratories. But even there, Bose-Einstein condensates are simulating the early universe, and quantum computers are simulating worm holes—tipping their hat to astrophysics!
So here are my picks for the top physics breakthroughs of 2023.
The Early Universe
The James Webb Space Telescope (JWST) has come through big on all of its promises! They said it would revolutionize the astrophysics of the early universe, and they were right. As of 2023, all astrophysics textbooks describing the early universe and the formation of galaxies are now obsolete, thanks to JWST.
Foremost among the discoveries is how fast the universe took up its current form. Galaxies condensed much earlier than expected, as did supermassive black holes. Everything that we thought took billions of years seem to have happened in only about one-tenth of that time (incredibly fast on cosmic time scales). The new JWST observations blow away the status quo on the early universe, and now the astrophysicists have to go back to the chalk board.
If LIGO and the first detection of gravitational waves was the huge breakthrough of 2015, detecting something so faint that it took a century to build an apparatus sensitive enough to detect them, then the newest observations of gravitational waves using galactic ripples presents a whole new level of gravitational wave physics.
By using the exquisitely precise timing of distant pulsars, astrophysicists have been able to detect a din of gravitational waves washing back and forth across the universe. These waves came from supermassive black hole mergers in the early universe. As the waves stretch and compress the space between us and distant pulsars, the arrival times of pulsar pulses detected at the Earth vary a tiny but measurable amount, haralding the passing of a gravitational wave.
This approach is a form of statistical optics in contrast to the original direct detection that was a form of interferometry. These are complimentary techniques in optics research, just as they will be complimentary forms of gravitational wave astronomy. Statistical optics (and fluctuation analysis) provides spectral density functions which can yield ensemble averages in the large N limit. This can answer questions about large ensembles that single interferometric detection cannot contribute to. Conversely, interferometric detection provides the details of individual events in ways that statistical optics cannot do. The two complimentary techniques, moving forward, will provide a much clearer picture of gravitational wave physics and the conditions in the universe that generate them.
Phosphorous on Enceladus
Planetary science is the close cousin to the more distant field of cosmology, but being close to home also makes it more immediate. The search for life outside the Earth stands as one of the greatest scientific quests of our day. We are almost certainly not alone in the universe, and life may be as close as Enceladus, the icy moon of Saturn.
Scientists have been studying data from the Cassini spacecraft that observed Saturn close-up for over a decade from 2004 to 2017. Enceladus has a subsurface liquid ocean that generates plumes of tiny ice crystals that erupt like geysers from fissures in the solid surface. The ocean remains liquid because of internal tidal heating caused by the large gravitational forces of Saturn.
The Cassini spacecraft flew through the plumes and analyzed their content using its Cosmic Dust Analyzer. While the ice crystals from Enceladus were already known to contain organic compounds, the science team discovered that they also contain phosphorous. This is the least abundant element within the molecules of life, but it is absolutely essential, providing the backbone chemistry of DNA as well as being a constituent of amino acids.
With this discovery, all the essential building blocks of life are known to exist on Enceladus, along with a liquid ocean that is likely to be in chemical contact with rocky minerals on the ocean floor, possibly providing the kind of environment that could promote the emergence of life on a planet other than Earth.
Simulating the Expanding Universe in a Bose-Einstein Condensate
Putting the universe under a microscope in a laboratory may have seemed a foolish dream, until a group at the University of Heidelberg did just that. It isn’t possible to make a real universe in the laboratory, but by adjusting the properties of an ultra-cold collection of atoms known as a Bose-Einstein condensate, the research group was able to create a type of local space whose internal metric has a curvature, like curved space-time. Furthermore, by controlling the inter-atomic interactions of the condensate with a magnetic field, they could cause the condensate to expand or contract, mimicking different scenarios for the evolution of our own universe. By adjusting the type of expansion that occurs, the scientists could create hypotheses about the geometry of the universe and test them experimentally, something that could never be done in our own universe. This could lead to new insights into the behavior of the early universe and the formation of its large-scale structure.
This is the only breakthrough I picked that is not related to astrophysics (although even this effect may have played a role in the very early universe).
Entanglement is one of the hottest topics in physics today (although the idea is 89 years old) because of the crucial role it plays in quantum information physics. The topic was awarded the 2022 Nobel Prize in Physics which went to John Clauser, Alain Aspect and Anton Zeilinger.
Direct observations of entanglement have been mostly restricted to optics (where entangled photons are easily created and detected) or molecular and atomic physics as well as in the solid state.
But entanglement eluded high-energy physics (which is quantum matter personified) until 2023 when the Atlas Collaboration at the LHC (Large Hadron Collider) in Geneva posted a manuscript on Arxiv that reported the first observation of entanglement in the decay products of a quark.
Fig. Thresholds for entanglement detection in decays from top quarks. Imagecredit.
Quarks interact so strongly (literally through the strong force), that entangled quarks experience very rapid decoherence, and entanglement effects virtually disappear in their decay products. However, top quarks decay so rapidly, that their entanglement properties can be transferred to their decay products, producing measurable effects in the downstream detection. This is what the Atlas team detected.
While this discovery won’t make quantum computers any better, it does open up a new perspective on high-energy particle interactions, and may even have contributed to the properties of the primordial soup during the Big Bang.
It may be hard to get excited about nothing … unless nothing is the whole ball game.
The only way we can really know what is, is by knowing what isn’t. Nothing is the backdrop against which we measure something. Experimentalists spend almost as much time doing control experiments, where nothing happens (or nothing is supposed to happen) as they spend measuring a phenomenon itself, the something.
Even the universe, full of so much something, came out of nothing during the Big Bang. And today the energy density of nothing, so-called Dark Energy, is blowing our universe apart, propelling it ever faster to a bitter cold end.
So here is a brief history of nothing, tracing how we have understood what it is, where it came from, and where is it today.
With sturdy shoulders, space stands opposing all its weight to nothingness. Where space is, there is being.
Friedrich Nietzsche
40,000 BCE – Cosmic Origins
This is a human history, about how we homo sapiens try to understand the natural world around us, so the first step on a history of nothing is the Big Bang of human consciousness that occurred sometime between 100,000 – 40,000 years ago. Some sort of collective phase transition happened in our thought process when we seem to have become aware of our own existence within the natural world. This time frame coincides with the beginning of representational art and ritual burial. This is also likely the time when human language skills reached their modern form, and when logical arguments–stories–first were told to explain our existence and origins.
Roughly two origin stories emerged from this time. One of these assumes that what is has always been, either continuously or cyclically. Buddhism and Hinduism are part of this tradition as are many of the origin philosophies of Indigenous North Americans. Another assumes that there was a beginning when everything came out of nothing. Abrahamic faiths (Let there be light!) subscribe to this creatio ex nihilo. What came before creation? Nothing!
500 BCE – Leucippus and Democritus Atomism
The Greek philosopher Leucippus and his student Democritus, living around 500 BCE, were the first to lay out the atomic theory in which the elements of substance were indivisible atoms of matter, and between the atoms of matter was void. The different materials around us were created by the different ways that these atoms collide and cluster together. Plato later adhered to this theory, developing ideas along these lines in his Timeaus.
300 BCE – Aristotle Vacuum
Aristotle is famous for arguing, in his Physics Book IV, Section 8, that nature abhors a vacuum (horror vacui) because any void would be immediately filled by the imposing matter surrounding it. He also argued more philosophically that nothing, by definition, cannot exist.
1644 – Rene Descartes Vortex Theory
Fast forward a millennia and a half, and theories of existence were finally achieving a level of sophistication that can be called “scientific”. Rene Descartes followed Aristotle’s views of the vacuum, but he extended it to the vacuum of space, filling it with an incompressible fluid in his Principles of Philosophy (1644). Just like water, laminar motion can only occur by shear, leading to vortices. Descartes was a better philosopher than mathematician, so it took Christian Huygens to apply mathematics to vortex motion to “explain” the gravitational effects of the solar system.
Otto von Guericke is one of those hidden gems of the history of science, a person who almost no-one remembers today, but who was far in advance of his own day. He was a powerful politician, holding the position of Burgomeister of the city of Magdeburg for more than 30 years, helping to rebuild it after it was sacked during the Thirty Years War. He was also a diplomat, playing a key role in the reorientation of power within the Holy Roman Empire. How he had free time is anyone’s guess, but he used it to pursue scientific interests that spanned from electrostatics to his invention of the vacuum pump.
With a succession of vacuum pumps, each better than the last, von Geuricke was like a kid in a toy factory, pumping the air out of anything he could find. In the process, he showed that a vacuum would extinguish a flame and could raise water in a tube.
His most famous demonstration was, of course, the Magdeburg sphere demonstration. In 1657 he fabricated two 20-inch hemispheres that he attached together with a vacuum seal and used his vacuum pump to evacuate the air from inside. He then attached chains from the hemispheres to a team of eight horses on each side, for a total of 16 horses, who were unable to separate the spheres. This dramatically demonstrated that air exerts a force on surfaces, and that Aristotle and Descartes were wrong—nature did allow a vacuum!
1667 – Isaac Newton Action at a Distance
When it came to the vacuum, Newton was agnostic. His universal theory of gravitation posited action at a distance, but the intervening medium played no direct role.
Nothing comes from nothing, Nothing ever could.
Rogers and Hammerstein, The Sound of Music
This would seem to say that Newton had nothing to say about the vacuum, but his other major work, his Optiks, established particles as the elements of light rays. Such light particles travelled easily through vacuum, so the particle theory of light came down on the empty side of space.
Statue of Isaac Newton by Sir Eduardo Paolozzi based on a painting by William Blake. Image Credit
1821 – Augustin Fresnel Luminiferous Aether
Today, we tend to think of Thomas Young as the chief proponent for the wave nature of light, going against the towering reputation of his own countryman Newton, and his courage and insights are admirable. But it was Augustin Fresnel who put mathematics to the theory. It was also Fresnel, working with his friend Francois Arago, who established that light waves are purely transverse.
For these contributions, Fresnel stands as one of the greatest physicists of the 1800’s. But his transverse light waves gave birth to one of the greatest red herrings of that century—the luminiferous aether. The argument went something like this, “if light is waves, then just as sound is oscillations of air, light must be oscillations of some medium that supports it – the luminiferous aether.” Arago searched for effects of this aether in his astronomical observations, but he didn’t see it, and Fresnel developed a theory of “partial aether drag” to account for Arago’s null measurement. Hippolyte Fizeau later confirmed the Fresnel “drag coefficient” in his famous measurement of the speed of light in moving water. (For the full story of Arago, Fresnel and Fizeau, see Chapter 2 of “Interference”. [1])
But the transverse character of light also required that this unknown medium must have some stiffness to it, like solids that support transverse elastic waves. This launched almost a century of alternative ideas of the aether that drew in such stellar actors as George Green, George Stokes and Augustin Cauchy with theories spanning from complete aether drag to zero aether drag with Fresnel’s partial aether drag somewhere in the middle.
1849 – Michael Faraday Field Theory
Micheal Faraday was one of the most intuitive physicists of the 1800’s. He worked by feel and mental images rather than by equations and proofs. He took nothing for granted, able to see what his experiments were telling him instead of looking only for what he expected.
This talent allowed him to see lines of force when he mapped out the magnetic field around a current-carrying wire. Physicists before him, including Ampere who developed a mathematical theory for the magnetic effects of a wire, thought only in terms of Newton’s action at a distance. All forces were central forces that acted in straight lines. Faraday’s experiments told him something different. The magnetic lines of force were circular, not straight. And they filled space. This realization led him to formulate his theory for the magnetic field.
Others at the time rejected this view, until William Thomson (the future Lord Kelvin) wrote a letter to Faraday in 1845 telling him that he had developed a mathematical theory for the field. He suggested that Faraday look for effects of fields on light, which Faraday found just one month later when he observed the rotation of the polarization of light when it propagated in a high-index material subject to a high magnetic field. This effect is now called Faraday Rotation and was one of the first experimental verifications of the direct effects of fields.
Nothing is more real than nothing.
Samuel Beckett
In 1949, Faraday stated his theory of fields in their strongest form, suggesting that fields in empty space were the repository of magnetic phenomena rather than magnets themselves [2]. He also proposed a theory of light in which the electric and magnetic fields induced each other in repeated succession without the need for a luminiferous aether.
1861 – James Clerk Maxwell Equations of Electromagnetism
James Clerk Maxwell pulled the various electric and magnetic phenomena together into a single grand theory, although the four succinct “Maxwell Equations” was condensed by Oliver Heaviside from Maxwell’s original 15 equations (written using Hamilton’s awkward quaternions) down to the 4 vector equations that we know and love today.
One of the most significant and most surprising thing to come out of Maxwell’s equations was the speed of electromagnetic waves that matched closely with the known speed of light, providing near certain proof that light was electromagnetic waves.
However, the propagation of electromagnetic waves in Maxwell’s theory did not rule out the existence of a supporting medium—the luminiferous aether. It was still not clear that fields could exist in a pure vacuum but might still be like the stress fields in solids.
Late in his life, just before he died, Maxwell pointed out that no measurement of relative speed through the aether performed on a moving Earth could see deviations that were linear in the speed of the Earth but instead would be second order. He considered that such second-order effects would be far to small ever to detect, but Albert Michelson had different ideas.
1887 – Albert Michelson Null Experiment
Albert Michelson was convinced of the existence of the luminiferous aether, and he was equally convinced that he could detect it. In 1880, working in the basement of the Potsdam Observatory outside Berlin, he operated his first interferometer in a search for evidence of the motion of the Earth through the aether. He had built the interferometer, what has come to be called a Michelson Interferometer, months earlier in the laboratory of Hermann von Helmholtz in the center of Berlin, but the footfalls of the horse carriages outside the building disturbed the measurements too much—Postdam was quieter.
But he could find no difference in his interference fringes as he oriented the arms of his interferometer parallel and orthogonal to the Earth’s motion. A simple calculation told him that his interferometer design should have been able to detect it—just barely—so the null experiment was a puzzle.
Seven years later, again in a basement (this time in a student dormitory at Western Reserve College in Cleveland, Ohio), Michelson repeated the experiment with an interferometer that was ten times more sensitive. He did this in collaboration with Edward Morley. But again, the results were null. There was no difference in the interference fringes regardless of which way he oriented his interferometer. Motion through the aether was undetectable.
(Michelson has a fascinating backstory, complete with firestorms (literally) and the Wild West and a moment when he was almost committed to an insane asylum against his will by a vengeful wife. To read all about this, see Chapter 4: After the Gold Rush in my recent book Interference (Oxford, 2023)).
The Michelson Morley experiment did not create the crisis in physics that it is sometimes credited with. They published their results, and the physics world took it in stride. Voigt and Fitzgerald and Lorentz and Poincaré toyed with various ideas to explain it away, but there had already been so many different models, from complete drag to no drag, that a few more theories just added to the bunch.
But they all had their heads in a haze. It took an unknown patent clerk in Switzerland to blow away the wisps and bring the problem into the crystal clear.
1905 – Albert Einstein Relativity
So much has been written about Albert Einstein’s “miracle year” of 1905 that it has lapsed into a form of physics mythology. Looking back, it seems like his own personal Big Bang, springing forth out of the vacuum. He published 5 papers that year, each one launching a new approach to physics on a bewildering breadth of problems from statistical mechanics to quantum physics, from electromagnetism to light … and of course, Special Relativity [3].
Whereas the others, Voigt and Fitzgerald and Lorentz and Poincaré, were trying to reconcile measurements of the speed of light in relative motion, Einstein just replaced all that musing with a simple postulate, his second postulate of relativity theory:
2. Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence …
Albert Einstein, Annalen der Physik, 1905
And the rest was just simple algebra—in complete agreement with Michelson’s null experiment, and with Fizeau’s measurement of the so-called Fresnel drag coefficient, while also leading to the famous E = mc2 and beyond.
There is no aether. Electromagnetic waves are self-supporting in vacuum—changing electric fields induce changing magnetic fields that induce, in turn, changing electric fields—and so it goes.
The vacuum is vacuum—nothing! Except that it isn’t. It is still full of things.
1931 – P. A. M Dirac Antimatter
The Dirac equation is the famous end-product of P. A. M. Dirac’s search for a relativistic form of the Schrödinger equation. It replaces the asymmetric use in Schrödinger’s form of a second spatial derivative and a first time derivative with Dirac’s form using only first derivatives that are compatible with relativistic transformations [4].
One of the immediate consequences of this equation is a solution that has negative energy. At first puzzling and hard to interpret [5], Dirac eventually hit on the amazing proposal that these negative energy states are real particles paired with ordinary particles. For instance, the negative energy state associated with the electron was an anti-electron, a particle with the same mass as the electron, but with positive charge. Furthermore, because the anti-electron has negative energy and the electron has positive energy, these two particles can annihilate and convert their mass energy into the energy of gamma rays. This audacious proposal was confirmed by the American physicist Carl Anderson who discovered the positron in 1932.
The existence of particles and anti-particles, combined with Heisenberg’s uncertainty principle, suggests that vacuum fluctuations can spontaneously produce electron-positron pairs that would then annihilate within a time related to the mass energy
Although this is an exceedingly short time (about 10-21 seconds), it means that the vacuum is not empty, but contains a frothing sea of particle-antiparticle pairs popping into and out of existence.
1938 – M. C. Escher Negative Space
Scientists are not the only ones who think about empty space. Artists, too, are deeply committed to a visual understanding of our world around us, and the uses of negative space in art dates back virtually to the first cave paintings. However, artists and art historians only talked explicitly in such terms since the 1930’s and 1940’s [6]. One of the best early examples of the interplay between positive and negative space was a print made by M. C. Escher in 1938 titled “Day and Night”.
1946 – Edward Purcell Modified Spontaneous Emission
In 1916 Einstein laid out the laws of photon emission and absorption using very simple arguments (his modus operandi) based on the principles of detailed balance. He discovered that light can be emitted either spontaneously or through stimulated emission (the basis of the laser) [7]. Once the nature of vacuum fluctuations was realized through the work of Dirac, spontaneous emission was understood more deeply as a form of stimulated emission caused by vacuum fluctuations. In the absence of vacuum fluctuations, spontaneous emission would be inhibited. Conversely, if vacuum fluctuations are enhanced, then spontaneous emission would be enhanced.
This effect was observed by Edward Purcell in 1946 through the observation of emission times of an atom in a RF cavity [8]. When the atomic transition was resonant with the cavity, spontaneous emission times were much faster. The Purcell enhancement factor is
where Q is the “Q” of the cavity, and V is the cavity volume. The physical basis of this effect is the modification of vacuum fluctuations by the cavity modes caused by interference effects. When cavity modes have constructive interference, then vacuum fluctuations are larger, and spontaneous emission is stimulated more quickly.
1948 – Hendrik Casimir Vacuum Force
Interference effects in a cavity affect the total energy of the system by excluding some modes which become inaccessible to vacuum fluctuations. This lowers the internal energy internal to a cavity relative to free space outside the cavity, resulting in a net “pressure” acting on the cavity. If two parallel plates are placed in close proximity, this would cause a force of attraction between them. The effect was predicted in 1948 by Hendrik Casimir [9], but it was not verified experimentally until 1997 by S. Lamoreaux at Yale University [10].
Two plates brought very close feel a pressure exerted by the higher vacuum energy density external to the cavity.
1949 – Shinichiro Tomonaga, Richard Feynman and Julian Schwinger QED
The physics of the vacuum in the years up to 1948 had been a hodge-podge of ad hoc theories that captured the qualitative aspects, and even some of the quantitative aspects of vacuum fluctuations, but a consistent theory was lacking until the work of Tomonaga in Japan, Feynman at Cornell and Schwinger at Harvard. Feynman and Schwinger both published their theory of quantum electrodynamics (QED) in 1949. They were actually scooped by Tomonaga, who had developed his theory earlier during WWII, but physics research in Japan had been cut off from the outside world. It was when Oppenheimer received a letter from Tomonaga in 1949 that the West became aware of his work. All three received the Nobel Prize for their work on QED in 1965. Precision tests of QED now make it one of the most accurately confirmed theories in physics.
Richard Feynman’s first “Feynman diagram”.
1964 – Peter Higgs and The Higgs
The Higgs particle, known as “The Higgs”, was the brain-child of Peter Higgs, Francois Englert and Gerald Guralnik in 1964. Higgs’ name became associated with the theory because of a response letter he wrote to an objection made about the theory. The Higg’s mechanism is spontaneous symmetry breaking in which a high-symmetry potential can lower its energy by distorting the field, arriving at a new minimum in the potential. This mechanism can allow the bosons that carry force to acquire mass (something the earlier Yang-Mills theory could not do).
Spontaneous symmetry breaking is a ubiquitous phenomenon in physics. It occurs in the solid state when crystals can lower their total energy by slightly distorting from a high symmetry to a low symmetry. It occurs in superconductors in the formation of Cooper pairs that carry supercurrents. And here it occurs in the Higgs field as the mechanism to imbues particles with mass .
Conceptual graph of a potential surface where the high symmetry potential is higher than when space is distorted to lower symmetry. Image Credit
The theory was mostly ignored for its first decade, but later became the core of theories of electroweak unification. The Large Hadron Collider (LHC) at Geneva was built to detect the boson, announced in 2012. Peter Higgs and Francois Englert were awarded the Nobel Prize in Physics in 2013, just one year after the discovery.
The Higgs field permeates all space, and distortions in this field around idealized massless point particles are observed as mass. In this way empty space becomes anything but.
1981 – Alan Guth Inflationary Big Bang
Problems arose in observational cosmology in the 1970’s when it was understood that parts of the observable universe that should have been causally disconnected were in thermal equilibrium. This could only be possible if the universe were much smaller near the very beginning. In January of 1981, Alan Guth, then at Cornell University, realized that a rapid expansion from an initial quantum fluctuation could be achieved if an initial “false vacuum” existed in a positive energy density state (negative vacuum pressure). Such a false vacuum could relax to the ordinary vacuum, causing a period of very rapid growth that Guth called “inflation”. Equilibrium would have been achieved prior to inflation, solving the observational problem.Therefore, the inflationary model posits a multiplicities of different types of “vacuum”, and once again, simple vacuum is not so simple.
Energy density as a function of a scalar variable. Quantum fluctuations create a “false vacuum” that can relax to “normal vacuum: by expanding rapidly. Image Credit
1998 – Saul Pearlmutter Dark Energy
Einstein didn’t make many mistakes, but in the early days of General Relativity he constructed a theoretical model of a “static” universe. A central parameter in Einstein’s model was something called the Cosmological Constant. By tuning it to balance gravitational collapse, he tuned the universe into a static Ithough unstable) state. But when Edwin Hubble showed that the universe was expanding, Einstein was proven incorrect. His Cosmological Constant was set to zero and was considered to be a rare blunder.
Fast forward to 1999, and the Supernova Cosmology Project, directed by Saul Pearlmutter, discovered that the expansion of the universe was accelerating. The simplest explanation was that Einstein had been right all along, or at least partially right, in that there was a non-zero Cosmological Constant. Not only is the universe not static, but it is literally blowing up. The physical origin of the Cosmological Constant is believed to be a form of energy density associated with the space of the universe. This “extra” energy density has been called “Dark Energy”, filling empty space.
The bottom line is that nothing, i.e., the vacuum, is far from nothing. It is filled with a froth of particles, and energy, and fields, and potentials, and broken symmetries, and negative pressures, and who knows what else as modern physics has been much ado about this so-called nothing, almost more than it has been about everything else.
[2] L. Peirce Williams in “Faraday, Michael.” Complete Dictionary of Scientific Biography, vol. 4, Charles Scribner’s Sons, 2008, pp. 527-540.
[3] A. Einstein, “On the electrodynamics of moving bodies,” Annalen Der Physik 17, 891-921 (1905).
[4] Dirac, P. A. M. (1928). “The Quantum Theory of the Electron”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 117 (778): 610–624.
[5] Dirac, P. A. M. (1930). “A Theory of Electrons and Protons”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 126 (801): 360–365.
[6] Nikolai M Kasak, Physical Art: Action of positive and negative space, (Rome, 1947/48) [2d part rev. in 1955 and 1956].
[7] A. Einstein, “Strahlungs-Emission un -Absorption nach der Quantentheorie,” Verh. Deutsch. Phys. Ges. 18, 318 (1916).
[8] Purcell, E. M. (1946-06-01). “Proceedings of the American Physical Society: Spontaneous Emission Probabilities at Ratio Frequencies”. Physical Review. American Physical Society (APS). 69 (11–12): 681.
[9] Casimir, H. B. G. (1948). “On the attraction between two perfectly conducting plates”. Proc. Kon. Ned. Akad. Wet. 51: 793.
[10] Lamoreaux, S. K. (1997). “Demonstration of the Casimir Force in the 0.6 to 6 μm Range”. Physical Review Letters. 78 (1): 5–8.
Fractals, those telescoping self-similar filigree meshes that marry mathematics and art, have become so mainstream, that they are even mentioned in the theme song of Disney’s 2013 mega-hit, Frozen.
My power flurries through the air into the ground My soul is spiraling in frozen fractals all around And one thought crystallizes like an icy blast I’m never going back, the past is in the past
Let it Go, by Idina Menzel (Frozen, Disney 2013)
But not all fractals are cut from the same cloth. Some are thin and some are fat. The thin ones are the ones we know best, adorning the cover of books and magazines. But the fat ones may be more common and may play important roles, such as in the stability of celestial orbits in a many-planet neighborhood, or in the stability and structure of Saturn’s rings.
To get a handle on fat fractals, we will start with a familiar thin one, the zero-measure Cantor set.
The Zero-Measure Cantor Set
The famous one-third Cantor set is often the first fractal that you encounter in any introduction to fractals. (See my blog on a short history of fractals.) It lives on a one-dimensional line, and its iterative construction is intuitive and simple.
Start with a long thin bar of unit length. Then remove the middle third, leaving the endpoints. This leaves two identical bars of one-third length each. Next, remove the open middle third of each of these, again leaving the endpoints, leaving behind section pairs of one-nineth length. Then repeat ad infinitum. The points of the line that remain–all those segment endpoints–are the Cantor set.
Fig. 1 Construction of the 1/3 Cantor set by removing 1/3 segments at each level, and leaving the endpoints of each segment. The resulting set is a dust of points with a fractal dimension D = ln(2)/ln(3) = 0.6309.
The Cantor set has a fractal dimension that is easily calculated by noting that at each stage there are two elements (N = 2) that divided by three in size (b = 3). The fractal dimension is then
It is easy to prove that the collection of points of the Cantor set have no length because all of the length was removed.
For instance, at the first level, one third of the length was removed. At the second level, two segments of one-nineth length were removed. At the third level, four segments of one-twenty-sevength length were removed, and so on. Mathematically, this is
The infinite series in the brackets is a binomial series with the simple solution
Therefore, all the length has been removed, and none is left to the Cantor set, which is simply a collection of all the endpoints of all the segments that were removed.
The Cantor set is said to have a Lebesgue measure of zero. It behaves as a dust of isolated points.
A close relative of the Cantor set is the Sierpinski Carpet which is the two-dimensional analog. It begins with a square of unit side, then the middle third is removed (one nineth of the three-by-three array of square of one-third side), and so on.
Fig. 2 A regular Sierpinski Carpet with fractal dimension D = ln(8)/ln(3) = 1.8928.
The resulting Sierpinski Carpet has zero Lebesgue measure, just like the Cantor dust, because all the area has been removed.
There are also random Sierpinski Carpets as the sub-squares are removed from random locations.
Fig. 3 A random Sierpinski Carpet with fractal dimension D = ln(8)/ln(3) = 1.8928.
These fractals are “thin”, so-called because they are dusts with zero measure.
But the construction was constructed just so, such that the sum over all the removed sub-lengths summed to unity. What if less material had been taken at each step? What happens?
Fat Fractals
Instead of taking one-third of the original length, take instead one-fourth. But keep the one-third scaling level-to-level, as for the original Cantor Set.
Fig. 4 A “fat” Cantor fractal constructed by removing 1/4 of a segment at each level instead of 1/3.
The total length removed is
Therefore, three fourths of the length was removed, leaving behind one fourth of the material. Not only that, but the material left behind is contiguous—solid lengths. At each level, a little bit of the original bar remains, and still remains at the next level and the next. Therefore, it is said to have a Lebesgue measure of unity. This construction leads to a “fat” fractal.
Fig. 5 Fat Cantor fractal showing the original Cantor 1/3 set (in black) and the extra contiguous segments (in red) that give the set a Lebesgue measure equal to one.
Looking at Fig. 5, it is clear that the original Cantor dust is still present as the black segments interspersed among the red parts of the bar that are contiguous. But when two sets are added that have different “dimensions”, then the combined set has the larger dimension of the two, which is one-dimensional in this case. The fat Cantor set is one dimensional. One can still study its scaling properties, leading to another type of dimension known as an exterior measure [1], but where do such fat fractals occur? Why do they matter?
One answer is that they lie within the oddly named “Arnold Tongues” that arise in the study of synchronization and resonance connected to the stability of the solar system and the safety of its inhabitants.
Arnold Tongues
The study of synchronization explores and explains how two or more non-identical oscillators can lock themselves onto a common shared oscillation. For two systems to synchronize requires autonomous oscillators (like planetary orbits) with a period-dependent interaction (like gravity). Such interactions are “resonant” when the periods of the two orbits are integer ratios of each other, like 1:2 or 2:3. Such resonances ensure that there is a periodic forcing caused by the interaction that is some multiple of the orbital period. Think of tapping a rotating bicycle wheel twice per cycle or three times per cycle. Even if you are a little off in your timing, you can lock the tire rotation rate to a multiple of your tapping frequency. But if you are too far off on your timing, then the wheel will turn independently of your tapping.
Because rational ratios of integers are plentiful, there can be an intricate interplay between locked frequencies and unlocked frequencies. When the rotation rate is close to a resonance, then the wheel can frequency-lock to the tapping. Plotting the regions where the wheel synchronizes or not as a function of the frequency ratio and also as a function of the strength of the tapping leads to one of the iconic images of nonlinear dynamics: the Arnold tongue diagram.
Fig. 6 Arnold tongue diagram, showing the regions of frequency locking (black) at rational resonances as a function of coupling strength. At unity coupling strength, the set outside frequency-locked regions is fractal with D = 0.87. For all smaller coupling, a set along a horizontal is a fat fractal with topological dimension D = 1. The white regions are “ergodic”, as the phase of the oscillator runs through all possible values.
The Arnold tongues in Fig. 6 are the frequency locked regions (black) as a function of frequency ratio and coupling strength g. The black regions correspond to rational ratios of frequencies. For g = 1, the set outside frequency-locked regions (the white regions are “ergodic”, as the phase of the oscillator runs through all possible values) is a thin fractal with D = 0.87. For g < 1, the sets outside the frequency locked regions along a horizontal (at constant g) are fat fractals with topological dimension D = 1. For fat fractals, the fractal dimension is irrelevant, and another scaling exponent takes on central importance.
The Lebesgue measure μ of the ergodic regions (the regions that are not frequency locked) is a function of the coupling strength varying from μ = 1 at g = 0 to μ = 0 at g = 1. When the pattern is coarse-grained at a scale ε, then the scaling of a fat fractal is
where β is the scaling exponent that characterizes the fat fractal.
From numerical studies [2] there is strong evidence that β = 2/3 for the fat fractals of Arnold Tongues.
The Rings of Saturn
Arnold Tongues arise in KAM theory on the stability of the solar system (See my blog on KAM and how number theory protects us from the chaos of the cosmos). Fortunately, Jupiter is the largest perturbation to Earth’s orbit, but its influence, while non-zero, is not enough to seriously affect our stability. However, there is a part of the solar system where rational resonances are not only large but dominant: Saturn’s rings.
Saturn’s rings are composed of dust and ice particles that orbit Saturn with a range of orbital periods. When one of these periods is a rational fraction of the orbital period of a moon, then a resonance condition is satisfied. Saturn has many moons, producing highly corrugated patterns in Saturn’s rings at rational resonances of the periods.
Fig. 7 A close up of Saturn’s rings shows a highly detailed set of bands. Particles at a given radius have a given period (set by Kepler’s third law). When the period of dust particles in the ring are an integer ratio of the period of a “shepherd moon”, then a resonance can drive density rings. [See image reference.]
The moons Janus and Epithemeus share an orbit around Saturn in a rare 1:1 resonance in which they swap positions every four years. Their combined gravity excites density ripples in Saturn’s rings, photographed by the Cassini spacecraft and shown in Fig. 8.
Fig. 8 Cassini spacecraft photograph of density ripples in Saturns rings caused by orbital resonance with the pair of moons Janus and Epithemeus.
One Canadian astronomy group converted the resonances of the moon Janus into a musical score to commenorate Cassini’s final dive into the planet Saturn in 2017. The Janus resonances are shown in Fig. 9 against the pattern of Saturn’s rings.
Fig. 7 Rational resonances for subrings of Saturn relative to its moon Janus.
Saturn’s rings, orbital resonances, Arnold tongues and fat fractals provide a beautiful example of the power of dynamics to create structure, and the primary role that structure plays in deciphering the physics of complex systems.
References:
[1] C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, “EXTERIOR DIMENSION OF FAT FRACTALS,” Physics Letters A 110, 1-4 (1985).
[2] R. E. Ecke, J. D. Farmer, and D. K. Umberger, “Scaling of the Arnold tongues,” Nonlinearity 2, 175-196 (1989).
Light is one of the most powerful manifestations of the forces of physics because it tells us about our reality. The interference of light, in particular, has led to the detection of exoplanets orbiting distant stars, discovery of the first gravitational waves, capture of images of black holes and much more. The stories behind the history of light and interference go to the heart of how scientists do what they do and what they often have to overcome to do it. These time-lines are organized along the chapter titles of the book Interference. They follow the path of theories of light from the first wave-particle debate, through the personal firestorms of Albert Michelson, to the discoveries of the present day in quantum information sciences.
Thomas Young was the ultimate dabbler, his interests and explorations ranged far and wide, from ancient egyptology to naval engineering, from physiology of perception to the physics of sound and light. Yet unlike most dabblers who accomplish little, he made original and seminal contributions to all these fields. Some have called him the “Last Man Who Knew Everything“.
Thomas Young. The Law of Interference.
Topics: The Law of Interference. The Rosetta Stone. Benjamin Thompson, Count Rumford. Royal Society. Christiaan Huygens. Pendulum Clocks. Icelandic Spar. Huygens’ Principle. Stellar Aberration. Speed of Light. Double-slit Experiment.
1629 – Huygens born (1629 – 1695)
1642 – Galileo dies, Newton born (1642 – 1727)
1655 – Huygens ring of Saturn
1657 – Huygens patents the pendulum clock
1666 – Newton prismatic colors
1666 – Huygens moves to Paris
1669 – Bartholin double refraction in Icelandic spar
1670 – Bartholinus polarization of light by crystals
1671 – Expedition to Hven by Picard and Rømer
1673 – James Gregory bird-feather diffraction grating
1801 – Young Theory of Light and Colours, three color mechanism (Bakerian Lecture), Young considers interference to cause the colored films, first estimates of the wavelengths of different colors
1802 – Young begins series of lecturs at the Royal Institution (Jan. 1802 – July 1803)
1802 – Young names the principle (Law) of interference
Augustin Fresnel was an intuitive genius whose talents were almost squandered on his job building roads and bridges in the backwaters of France until he was discovered and rescued by Francois Arago.
Topics: Particles versus Waves. Malus and Polarization. Agustin Fresnel. Francois Arago. Diffraction. Daniel Bernoulli. The Principle of Superposition. Joseph Fourier. Transverse Light Waves.
1665 – Grimaldi diffraction bands outside shadow
1673 – James Gregory bird-feather diffraction grating
There is no question that Francois Arago was a swashbuckler. His life’s story reads like an adventure novel as he went from being marooned in hostile lands early in his career to becoming prime minister of France after the 1848 revolutions swept across Europe.
Topics: The Birth of Interferometry. Snell’s Law. Fresnel and Arago. The First Interferometer. Fizeau and Foucault. The Speed of Light. Ether Drag. Jamin Interferometer.
No name is more closely connected to interferometry than that of Albert Michelson. He succeeded, sometimes at great personal cost, in launching interferometric metrology as one of the most important tools used by scientists today.
Albert A. Michelson, 1907 Nobel Prize. Image Credit.
Topics: The Trials of Albert Michelson. Hermann von Helmholtz. Michelson and Morley. Fabry and Perot.
1810 – Arago search for ether drag
1813 – Fraunhofer dark lines in Sun spectrum
1813 – Faraday begins at Royal Institution
1820 – Oersted discovers electromagnetism
1821 – Faraday electromagnetic phenomena
1827 – Green mathematical analysis of electricity and magnetism
1830 – Cauchy ether as elastic solid
1831 – Faraday electromagnetic induction
1831 – Cauchy ether drag
1831 – Maxwell born
1831 – Faraday electromagnetic induction
1836 – Cauchy’s second theory of the ether
1838 – Green theory of the ether
1839 – Hamilton group velocity
1839 – MacCullagh properties of rotational ether
1839 – Cauchy ether with negative compressibility
1841 – Maxwell entered Edinburgh Academy (age 10) met P. G. Tait
1842 – Doppler effect
1845 – Faraday effect (magneto-optic rotation)
1846 – Stokes’ viscoelastic theory of the ether
1847 – Maxwell entered Edinburgh University
1850 – Maxwell at Cambridge, studied under Hopkins, also knew Stokes and Whewell
1852 – Michelson born Strelno, Prussia
1854 – Maxwell wins the Smith’s Prize (Stokes’ theorem was one of the problems)
1855 – Michelson’s immigrate to San Francisco through Panama Canal
Learning from his attempts to measure the speed of light through the ether, Michelson realized that the partial coherence of light from astronomical sources could be used to measure their sizes. His first measurements using the Michelson Stellar Interferometer launched a major subfield of astronomy that is one of the most active today.
R Hanbury Brown
Topics: Measuring the Stars. Astrometry. Moons of Jupiter. Schwarzschild. Betelgeuse. Michelson Stellar Interferometer. Banbury Brown Twiss. Sirius. Adaptive Optics.
1838 – Bessel stellar parallax measurement with Fraunhofer telescope
1868 – Fizeau proposes stellar interferometry
1873 – Stephan implements Fizeau’s stellar interferometer on Sirius, sees fringes
1880 – Michelson Idea for second-order measurement of relative motion against ether
1880 – 1882 Michelson Studies in Europe (Helmholtz in Berlin, Quincke in Heidelberg, Cornu, Mascart and Lippman in Paris)
1881 – Michelson Measurement at Potsdam with funds from Alexander Graham Bell
1881 – Michelson Resigned from active duty in the Navy
1883 – Michelson Joined Case School of Applied Science
1889 – Michelson moved to Clark University at Worcester
Stellar interferometry is opening new vistas of astronomy, exploring the wildest occupants of our universe, from colliding black holes half-way across the universe (LIGO) to images of neighboring black holes (EHT) to exoplanets near Earth that may harbor life.
Image of the supermassive black hole in M87 from Event Horizon Telescope.
Topics: Gravitational Waves, Black Holes and the Search for Exoplanets. Nulling Interferometer. Event Horizon Telescope. M87 Black Hole. Long Baseline Interferometry. LIGO.
1947 – Virgo A radio source identified as M87
1953 – Horace W. Babcock proposes adaptive optics (AO)
From the astronomically large dimensions of outer space to the microscopically small dimensions of inner space, optical interference pushes the resolution limits of imaging.
Topics: Diffraction and Interference. Joseph Fraunhofer. Diffraction Gratings. Henry Rowland. Carl Zeiss. Ernst Abbe. Phase-contrast Microscopy. Super-resolution Micrscopes. Structured Illumination.
The coherence of laser light is like a brilliant jewel that sparkles in the darkness, illuminating life, probing science and projecting holograms in virtual worlds.
What is the image of one photon interfering? Better yet, what is the image of two photons interfering? The answer to this crucial question laid the foundation for quantum communication.
Topics: The Beginnings of Quantum Communication. EPR paradox. Entanglement. David Bohm. John Bell. The Bell Inequalities. Leonard Mandel. Single-photon Interferometry. HOM Interferometer. Two-photon Fringes. Quantum cryptography. Quantum Teleportation.
1900 – Planck (1901). “Law of energy distribution in normal spectra.” [1]
There is almost no technical advantage better than having exponential resources at hand. The exponential resources of quantum interference provide that advantage to quantum computing which is poised to usher in a new era of quantum information science and technology.
David Deutsch.
Topics: Interferometric Computing. David Deutsch. Quantum Algorithm. Peter Shor. Prime Factorization. Quantum Logic Gates. Linear Optical Quantum Computing. Boson Sampling. Quantum Computational Advantage.
1980 – Paul Benioff describes possibility of quantum computer
[10] B. R. Mollow, R. J. Glauber: Phys. Rev. 160, 1097 (1967); 162, 1256 (1967)
[11] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, ” Proposed experiment to test local hidden-variable theories,” Physical Review Letters, vol. 23, no. 15, pp. 880-&, (1969)
[15] R. Ghosh and L. Mandel, “Observation of nonclassical effects in the interference of 2 photons,” Physical Review Letters, vol. 59, no. 17, pp. 1903-1905, Oct (1987)
[16] C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between 2 photons by interference,” Physical Review Letters, vol. 59, no. 18, pp. 2044-2046, Nov (1987)
[18] D. Deutsch, “QUANTUM-THEORY, THE CHURCH-TURING PRINCIPLE AND THE UNIVERSAL QUANTUM COMPUTER,” Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, vol. 400, no. 1818, pp. 97-117, (1985)
[19] P. W. Shor, “ALGORITHMS FOR QUANTUM COMPUTATION – DISCRETE LOGARITHMS AND FACTORING,” in 35th Annual Symposium on Foundations of Computer Science, Proceedings, S. Goldwasser Ed., (Annual Symposium on Foundations of Computer Science, 1994, pp. 124-134.
[20] F. Arute et al., “Quantum supremacy using a programmable superconducting processor,” Nature, vol. 574, no. 7779, pp. 505-+, Oct 24 (2019)
[21] H.-S. Zhong et al., “Quantum computational advantage using photons,” Science, vol. 370, no. 6523, p. 1460, (2020)
Further Reading: The History of Light and Interference (2023)
The first step on the road to Einstein’s relativity was taken a hundred years earlier by an ironic rebel of physics—Augustin Fresnel. His radical (at the time) wave theory of light was so successful, especially the proof that it must be composed of transverse waves, that he was single-handedly responsible for creating the irksome luminiferous aether that would haunt physicists for the next century. It was only when Einstein combined the work of Fresnel with that of Hippolyte Fizeau that the aether was ultimately banished.
Augustin Fresnel: Ironic Rebel of Physics
Augustin Fresnel was an odd genius who struggled to find his place in the technical hierarchies of France. After graduating from the Ecole Polytechnique, Fresnel was assigned a mindless job overseeing the building of roads and bridges in the boondocks of France—work he hated. To keep himself from going mad, he toyed with physics in his spare time, and he stumbled on inconsistencies in Newton’s particulate theory of light that Laplace, a leader of the French scientific community, embraced as if it were revealed truth .
The final irony is that Einstein used Fresnel’s theoretical coefficient and Fizeau’s measurements—that had introduced aether drag in the first place—to show that there was no aether.
Fresnel rebelled, realizing that effects of diffraction could be explained if light were made of waves. He wrote up an initial outline of his new wave theory of light, but he could get no one to listen, until Francois Arago heard of it. Arago was having his own doubts about the particle theory of light based on his experiments on stellar aberration.
Augustin Fresnel and Francois Arago (circa 1818)
Stellar Aberration and the Fresnel Drag Coefficient
Stellar aberration had been explained by James Bradley in 1729 as the effect of the motion of the Earth relative to the motion of light “particles” coming from a star. The Earth’s motion made it look like the star was tilted at a very small angle (see my previous blog). That explanation had worked fine for nearly a hundred years, but then around 1810 Francois Arago at the Paris Observatory made extremely precise measurements of stellar aberration while placing finely ground glass prisms in front of his telescope. According to Snell’s law of refraction, which depended on the velocity of the light particles, the refraction angle should have been different at different times of the year when the Earth was moving one way or another relative to the speed of the light particles. But to high precision the effect was absent. Arago began to question the particle theory of light. When he heard about Fresnel’s work on the wave theory, he arranged a meeting, encouraging Fresnel to continue his work.
But at just this moment, in March of 1815, Napoleon returned from exile in Elba and began his march on Paris with a swelling army of soldiers who flocked to him. Fresnel rebelled again, joining a royalist militia to oppose Napoleon’s return. Napoleon won, but so did Fresnel, who was ironically placed under house arrest, which was like heaven to him. It freed him from building roads and bridges, giving him free time to do optics experiments in his mother’s house to support his growing theoretical work on the wave nature of light.
Arago convinced the authorities to allow Fresnel to come to Paris, where the two began experiments on diffraction and interference. By using polarizers to control the polarization of the interfering light paths, they concluded that light must be composed of transverse waves.
This brilliant insight was then followed by one of the great tragedies of science—waves needed a medium within which to propagate, so Fresnel conceived of the luminiferous aether to support it. Worse, the transverse properties of light required the aether to have a form of crystalline stiffness.
How could moving objects, like the Earth orbiting the sun, travel through such an aether without resistance? This was a serious problem for physics. One solution was that the aether was entrained by matter, so that as matter moved, the aether was dragged along with it. That solved the resistance problem, but it raised others, because it couldn’t explain Arago’s refraction measurements of aberration.
Fresnel realized that Arago’s null results could be explained if aether was only partially dragged along by matter. For instance, in the glass prisms used by Arago, the fraction of the aether being dragged along by the moving glass versus at rest would depend on the refractive index n of the glass. The speed of light in moving glass would then be
where c is the speed of light through stationary aether, vg is the speed of the glass prism through the stationary aether, and V is the speed of light in the moving glass. The first term in the expression is the ordinary definition of the speed of light in stationary matter with the refractive index. The second term is called the Fresnel drag coefficient which he communicated to Arago in a letter in 1818. Even at the high speed of the Earth moving around the sun, this second term is a correction of only about one part in ten thousand. It explained Arago’s null results for stellar aberration, but it was not possible to measure it directly in the laboratory at that time.
Fizeau’s Moving Water Experiment
Hippolyte Fizeau has the distinction of being the first to measure the speed of light directly in an Earth-bound experiment. All previous measurements had been astronomical. The story of his ingenious use of a chopper wheel and long-distance reflecting mirrors placed across the city of Paris in 1849 can be found in Chapter 3 of Interference. However, two years later he completed an experiment that few at the time noticed but which had a much more profound impact on the history of physics.
Hippolyte Fizeau
In 1851, Fizeau modified an Arago interferometer to pass two interfering light beams along pipes of moving water. The goal of the experiment was to measure the aether drag coefficient directly and to test Fresnel’s theory of partial aether drag. The interferometer allowed Fizeau to measure the speed of light in moving water relative to the speed of light in stationary water. The results of the experiment confirmed Fresnel’s drag coefficient to high accuracy, which seemed to confirm the partial drag of aether by moving matter.
Fizeau’s 1851 measurement of the speed of light in water using a modified Arago interferometer. (Reprinted from Chapter 2: Interference.)
This result stood for thirty years, presenting its own challenges for physicist exploring theories of the aether. The sophistication of interferometry improved over that time, and in 1881 Albert Michelson used his newly-invented interferometer to measure the speed of the Earth through the aether. He performed the experiment in the Potsdam Observatory outside Berlin, Germany, and found the opposite result of complete aether drag, contradicting Fizeau’s experiment. Later, after he began collaborating with Edwin Morley at Case and Western Reserve Colleges in Cleveland, Ohio, the two repeated Fizeau’s experiment to even better precision, finding once again Fresnel’s drag coefficient, followed by their own experiment, known now as “the Michelson-Morley Experiment” in 1887, that found no effect of the Earth’s movement through the aether.
The two experiments—Fizeau’s measurement of the Fresnel drag coefficient, and Michelson’s null measurement of the Earth’s motion—were in direct contradiction with each other. Based on the theory of the aether, they could not both be true.
But where to go from there? For the next 15 years, there were numerous attempts to put bandages on the aether theory, from Fitzgerald’s contraction to Lorenz’ transformations, but it all seemed like kludges built on top of kludges. None of it was elegant—until Einstein had his crucial insight.
Einstein’s Insight
While all the other top physicists at the time were trying to save the aether, taking its real existence as a fact of Nature to be reconciled with experiment, Einstein took the opposite approach—he assumed that the aether did not exist and began looking for what the experimental consequences would be.
From the days of Galileo, it was known that measured speeds depended on the frame of reference. This is why a knife dropped by a sailor climbing the mast of a moving ship strikes at the base of the mast, falling in a straight line in the sailor’s frame of reference, but an observer on the shore sees the knife making an arc—velocities of relative motion must add. But physicists had over-generalized this result and tried to apply it to light—Arago, Fresnel, Fizeau, Michelson, Lorenz—they were all locked in a mindset.
Einstein stepped outside that mindset and asked what would happen if all relatively moving observers measured the same value for the speed of light, regardless of their relative motion. It was just a little algebra to find that the way to add the speed of light c to the speed of a moving reference frame vref was
where the numerator was the usual Galilean relativity velocity addition, and the denominator was required to enforce the constancy of observed light speeds. Therefore, adding the speed of light to the speed of a moving reference frame gives back simply the speed of light.
Generalizing this equation for general velocity addition between moving frames gives
where u is now the speed of some moving object being added the the speed of a reference frame, and vobs is the “net” speed observed by some “external” observer . This is Einstein’s famous equation for relativistic velocity addition (see pg. 12 of the English translation). It ensures that all observers with differently moving frames all measure the same speed of light, while also predicting that no velocities for objects can ever exceed the speed of light.
This last fact is a consequence, not an assumption, as can be seen by letting the reference speed vref increase towards the speed of light so that vref ≈ c, then
so that the speed of an object launched in the forward direction from a reference frame moving near the speed of light is still observed to be no faster than the speed of light
All of this, so far, is theoretical. Einstein then looked to find some experimental verification of his new theory of relativistic velocity addition, and he thought of the Fizeau experimental measurement of the speed of light in moving water. Applying his new velocity addition formula to the Fizeau experiment, he set vref = vwater and u = c/n and found
The second term in the denominator is much smaller that unity and is expanded in a Taylor’s expansion
The last line is exactly the Fresnel drag coefficient!
Therefore, Fizeau, half a century before, in 1851, had already provided experimental verification of Einstein’s new theory for relativistic velocity addition! It wasn’t aether drag at all—it was relativistic velocity addition.
From this point onward, Einstein followed consequence after inexorable consequence, constructing what is now called his theory of Special Relativity, complete with relativistic transformations of time and space and energy and matter—all following from a simple postulate of the constancy of the speed of light and the prescription for the addition of velocities.
The final irony is that Einstein used Fresnel’s theoretical coefficient and Fizeau’s measurements, that had established aether drag in the first place, as the proof he needed to show that there was no aether. It was all just how you looked at it.
• The history behind Einstein’s use of relativistic velocity addition is given in: A. Pais, Subtle is the Lord: The Science and the Life of Albert Einstein (Oxford University Press, 2005).
The Earth races around the sun with remarkable speed—at over one hundred thousand kilometers per hour on its yearly track. This is about 0.01% of the speed of light—a small but non-negligible amount for which careful measurement might show the very first evidence of relativistic effects. How big is this effect and how do you measure it? One answer is the aberration of starlight, which is the slight deviation in the apparent position of stars caused by the linear speed of the Earth around the sun.
This is not parallax, which is caused the the changing position of the Earth around the sun. Ever since Copernicus, astronomers had been searching for parallax, which would give some indication how far away stars were. It was an important question, because the answer would say something about how big the universe was. But in the process of looking for parallax, astronomers found something else, something about 50 times bigger—aberration.
Aberration is the effect of the transverse speed of the Earth added to the speed of light coming from a star. For instance, this effect on the apparent location of stars in the sky is a simple calculation of the arctangent of 0.01%, which is an angle of about 20 seconds of arc, or about 40 seconds when comparing two angles 6 months apart. This was a bit bigger than the accuracy of astronomical measurements at the time when Jean Picard travelled from Paris to Denmark in 1671 to visit the ruins of the old observatory of Tycho Brahe at Uranibourg.
Fig. 1 Stellar parallax is the change in apparent positions of a star caused by the change in the Earth’s position as it orbits the sun. If the change in angle (θ) could be measured, then based on Newton’s theory of gravitation that gives the radius of the Earth’s orbit (R), the distance to the star (L) could be found.
Jean Picard at Uranibourg
Fig. 2 A view of Tycho Brahe’s Uranibourg astronomical observatory in Hven, Denmark. Tycho had to abandon it near the end of his life when a new king thought he was performing witchcraft.
Jean Picard went to Uranibourg originally in 1671, and during subsequent years, to measure the eclipses of the moons of Jupiter to determine longitude at sea—an idea first proposed by Galileo. When visiting Copenhagen, before heading out to the old observatory, Picard secured the services of an as yet unknown astronomer by the name of Ole Rømer. While at Uranibourg, Picard and Rømer made their required measurements of the eclipses of the moons of Jupiter, but with extra observation hours, Picard also made measurements of the positions of selected stars, such as Polaris, the North Star. His very precise measurements allowed him to track a tiny yearly shift, an aberration, in position by about 40 seconds of arc. At the time (before Rømer’s great insight about the finite speed of light—see Chapter 1 of Interference (Oxford, 2023)), the speed of light was thought to be either infinite or unmeasurably fast, so Picard thought that this shift was the long-sought effect of stellar parallax that would serve as a way to measure the distance to the stars. However, the direction of the shift of Polaris was completely wrong if it were caused by parallax, and Picard’s stellar aberration remained a mystery.
Fig. 3 Jean Picard (left) and his modern name-sake (right).
Samuel Molyneux and Murder in Kew
In 1725, the amateur Irish astronomer Samuel Molyneux (1689 – 1828) decided that the tools of astronomy had improved to the point that the question of parallax could be answered. He enlisted the help of an instrument maker outside London to install a 24-foot zenith sector (a telescope that points vertically upwards) at his home in Kew. Molyneux was an independently wealthy politician (he had married the first daughter of the second Earl of Essex) who sat in the British House of Commons, and he was also secretary to the Prince of Wales (the future George II). Because his political activities made demands on his time, he looked for assistance with his observations and invited James Bradley (1693 – 1762), the newly installed Savilian Professor of Astronomy at Oxford University, to join him in his search.
Fig. 4 James Bradley.
James Bradley was a rising star in the scientific circles of England. He came from a modest background but had the good fortune that his mother’s brother, James Pound, was a noted amateur astronomer who had set up a small observatory at his rectory in Wanstead. Bradley showed an early interest in astronomy, and Pound encouraged him, helping with the finances of his education that took him to degrees at Baliol College at Oxford. Even more fortunate was the fact that Pound’s close friend was the Astronomer Royal Edmund Halley, who also took a special interest in Bradley. With Halley’s encouragement, Bradley made important measurements of Mars and several nebulae, demonstrating an ability to work with great accuracy. Halley was impressed and nominated Bradley to the Royal Society in 1718, telling everyone that Bradley was destined to be one of the great astronomers of his time.
Molyneux must have sensed immediately that he had chosen wisely by selecting Bradley to help him with the parallax measurements. Bradley was capable of exceedingly precise work and was fluent mathematically with the geometric complexities of celestial orbits. Fastening the large zenith sector to the chimney of the house gave the apparatus great stability, and in December of 1725 they commenced observations of Gamma Draconis as it passed directly overhead. Because of the accuracy of the sector, they quickly observed a deviation in the star’s position, but the deviation was in the wrong direction, just as Picard had observed. They continued to make observations over two years, obtaining a detailed map of a yearly wobble in the star’s position as it changed angle by 40 seconds of arc (about one percent of a degree) over six months.
When Molyneux was appointed Lord of the Admiralty in 1727, as well as becoming a member of the Irish Parliament (representing Dublin University), he had little time to continue with the observations of Gamma Draconis. He helped Bradley set up a Zenith sector telescope at Bradley’s uncle’s observatory in Wanstead that had a wider field of view to observe more stars, and then he left the project to his friend. A few months later, before either he or Bradley had understood the cause of the stellar aberration, Molyneux collapsed while in the House of Commons and was carried back to his house. One of Molyneux’s many friends was the court anatomist Nathaniel St. André who attended to him over the next several days as he declined and died. St. André was already notorious for roles he had played in several public hoaxes, and on the night of his friend’s death, before the body had grown cold, he eloped with Molyneux’s wife, raising accusations of murder (that could never be proven).
James Bradley and the Light Wind
Over the following year, Bradley observed aberrations in several stars, all of them displaying the same yearly wobble of about 40 seconds of arc. This common behavior of numerous stars demanded a common explanation, something they all shared. It is said that the answer came to Bradley while he was boating on the Thames. The story may be apocryphal, but he apparently noticed the banner fluttering downwind at the top of the mast, and after the boat came about, the banner pointed in a new direction. The wind direction itself had not altered, but the motion of the boat relative to the wind had changed. Light at that time was considered to be made of a flux of corpuscles, like a gentle wind of particles. As the Earth orbited the Sun, its motion relative to this wind would change periodically with the seasons, and the apparent direction of the star would shift a little as a result.
Fig. 5 Principle of stellar aberration. On the left is the rest frame of the star positioned directly overhead as a moving telescope tube must be slightly tilted at an angle (equal to the arctangent of the ratio of the Earth’s speed to the speed of light–greatly exaggerated in the figure) to allow the light to pass through it. On the right is the rest frame of the telescope in which the angular position of the star appears shifted.
Bradley shared his observations and his explanation in a letter to Halley that was read before the Royal Society in January of 1729. Based on his observations, he calculated the speed of light to be about ten thousand times faster than the speed of the Earth in its orbit around the Sun. At that speed, it should take light eight minutes and twelve seconds to travel from the Sun to the Earth (the actual number is eight minutes and 19 seconds). This number was accurate to within a percent of the true value compared with the estimates made by Huygens from the eclipses of the moons of Jupiter that were in error by 27 percent. In addition, because he was unable to discern any effect of parallax in the stellar motions, Bradley was able to place a limit on how far the distant stars must be, more than 100,000 times farther the distance of the Earth from the Sun, which was much farther away than any had previously expected. In January of 1729 the size of the universe suddenly jumped to an incomprehensibly large scale.
Bradley’s explanation of the aberration of starlight was simple and matched observations with good quantitative accuracy. The particle nature of light made it like a wind, or a current, and the motion of the Earth was just a case of Galilean relativity that any freshman physics student can calculate. At first there seemed to be no controversy or difficulties with this interpretation. However, an obscure paper published in 1784 by an obscure English natural philosopher named John Michell (the first person to conceive of a “dark star”) opened a Pandora’s box that launched the crisis of the luminiferous ether and the eventual triumph of Einstein’s theory of Relativity (see Chapter 3 of Interference (Oxford, 2023)), .
Galileo Unbound 
